rsin A (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 6.2s

Alternatives: 22

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + b));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((a + b))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / Math.cos((a + b));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((a + b))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((a + b));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + b));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((a + b))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / Math.cos((a + b));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((a + b))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((a + b));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* (cos b) (cos a)))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(sin(a), -sin(b), (cos(b) * cos(a)));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * cos(a))))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   11. lower-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   6. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

   14. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

   15. lift-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

   6. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

   8. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

   10. *-commutative N/A

       \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

   14. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

   16. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

   17. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

   18. lift-*.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos b) (cos a) (* (- (sin b)) (sin a)))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(b), cos(a), (-sin(b) * sin(a)));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), Float64(Float64(-sin(b)) * sin(a))))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   11. lower-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   6. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

   14. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

   15. lift-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* (sin b) r) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))

C

double code(double r, double a, double b) {
	return (sin(b) * r) / ((cos(b) * cos(a)) - (sin(b) * sin(a)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) * r) / ((cos(b) * cos(a)) - (sin(b) * sin(a)))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) * r) / ((Math.cos(b) * Math.cos(a)) - (Math.sin(b) * Math.sin(a)));
}

Python

def code(r, a, b):
	return (math.sin(b) * r) / ((math.cos(b) * math.cos(a)) - (math.sin(b) * math.sin(a)))

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / Float64(Float64(cos(b) * cos(a)) - Float64(sin(b) * sin(a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (sin(b) * r) / ((cos(b) * cos(a)) - (sin(b) * sin(a)));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   11. lower-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   2. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   5. lift-*.f64 99.5

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
6. Add Preprocessing

Alternative 4: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -105000000000:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.0001984126984126984, -0.008333333333333333\right), b \cdot b, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -105000000000.0)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 4.0)
       (/
        t_0
        (fma
         (cos b)
         (cos a)
         (*
          (*
           (fma
            (fma
             (fma (* b b) 0.0001984126984126984 -0.008333333333333333)
             (* b b)
             0.16666666666666666)
            (* b b)
            -1.0)
           b)
          (sin a))))
       (/ t_0 (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -105000000000.0) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 4.0) {
		tmp = t_0 / fma(cos(b), cos(a), ((fma(fma(fma((b * b), 0.0001984126984126984, -0.008333333333333333), (b * b), 0.16666666666666666), (b * b), -1.0) * b) * sin(a)));
	} else {
		tmp = t_0 / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -105000000000.0)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 4.0)
		tmp = Float64(t_0 / fma(cos(b), cos(a), Float64(Float64(fma(fma(fma(Float64(b * b), 0.0001984126984126984, -0.008333333333333333), Float64(b * b), 0.16666666666666666), Float64(b * b), -1.0) * b) * sin(a))));
	else
		tmp = Float64(t_0 / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000000000.0], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.0], N[(t$95$0 / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e11

   1. Initial program 53.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.7

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.7%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 56.7

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 56.7

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -1.05e11 < b < 4

   1. Initial program 98.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(b \cdot \left({b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{1}{5040} \cdot {b}^{2} - \frac{1}{120}\right)\right) - 1\right)\right)} \cdot \sin a\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left({b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{1}{5040} \cdot {b}^{2} - \frac{1}{120}\right)\right) - 1\right) \cdot \color{blue}{b}\right) \cdot \sin a\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left({b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{1}{5040} \cdot {b}^{2} - \frac{1}{120}\right)\right) - 1\right) \cdot \color{blue}{b}\right) \cdot \sin a\right)} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.0001984126984126984, -0.008333333333333333\right), b \cdot b, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right)} \cdot \sin a\right)} \]

   if 4 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.5:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 0.62:\\ \;\;\;\;\frac{r \cdot t\_0}{\cos b \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma
            (fma (* b b) -0.0001984126984126984 0.008333333333333333)
            (* b b)
            -0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.5)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 0.62)
       (/ (* r t_0) (- (* (cos b) (cos a)) (* t_0 (sin a))))
       (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma(fma((b * b), -0.0001984126984126984, 0.008333333333333333), (b * b), -0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.5) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 0.62) {
		tmp = (r * t_0) / ((cos(b) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = (r * sin(b)) / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333), Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.5)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 0.62)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(cos(b) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.5], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.62], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.5

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 57.0

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 57.0

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 57.0

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -0.5 < b < 0.619999999999999996

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{5040}, \frac{1}{120}\right), b \cdot b, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{5040}, \frac{1}{120}\right), b \cdot b, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{5040}, \frac{1}{120}\right), b \cdot b, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

   9. Applied rewrites 99.7%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   if 0.619999999999999996 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.1%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -105000000000:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 7:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -105000000000.0)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 7.0)
       (/
        t_0
        (fma
         (cos b)
         (cos a)
         (*
          (*
           (fma
            (fma (* b b) -0.008333333333333333 0.16666666666666666)
            (* b b)
            -1.0)
           b)
          (sin a))))
       (/ t_0 (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -105000000000.0) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 7.0) {
		tmp = t_0 / fma(cos(b), cos(a), ((fma(fma((b * b), -0.008333333333333333, 0.16666666666666666), (b * b), -1.0) * b) * sin(a)));
	} else {
		tmp = t_0 / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -105000000000.0)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 7.0)
		tmp = Float64(t_0 / fma(cos(b), cos(a), Float64(Float64(fma(fma(Float64(b * b), -0.008333333333333333, 0.16666666666666666), Float64(b * b), -1.0) * b) * sin(a))));
	else
		tmp = Float64(t_0 / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000000000.0], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.0], N[(t$95$0 / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e11

   1. Initial program 53.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.7

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.7%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 56.7

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 56.7

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -1.05e11 < b < 7

   1. Initial program 98.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(b \cdot \left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right)\right)} \cdot \sin a\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right) \cdot \color{blue}{b}\right) \cdot \sin a\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right) \cdot \color{blue}{b}\right) \cdot \sin a\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot b\right) \cdot \sin a\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left(\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) \cdot {b}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot b\right) \cdot \sin a\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\left(\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) \cdot {b}^{2} + -1\right) \cdot b\right) \cdot \sin a\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}, {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\frac{-1}{120} \cdot {b}^{2} + \frac{1}{6}, {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left({b}^{2} \cdot \frac{-1}{120} + \frac{1}{6}, {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{-1}{120}, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{120}, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{120}, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{120}, \frac{1}{6}\right), b \cdot b, -1\right) \cdot b\right) \cdot \sin a\right)} \]

      13. lift-*.f64 99.0

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \sin a\right)} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right)} \cdot \sin a\right)} \]

   if 7 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -105000000000:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 7:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, b \cdot b, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b, \cos b \cdot \cos a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -105000000000.0)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 7.0)
       (/
        t_0
        (fma
         (sin a)
         (*
          (fma
           (fma -0.008333333333333333 (* b b) 0.16666666666666666)
           (* b b)
           -1.0)
          b)
         (* (cos b) (cos a))))
       (/ t_0 (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -105000000000.0) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 7.0) {
		tmp = t_0 / fma(sin(a), (fma(fma(-0.008333333333333333, (b * b), 0.16666666666666666), (b * b), -1.0) * b), (cos(b) * cos(a)));
	} else {
		tmp = t_0 / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -105000000000.0)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 7.0)
		tmp = Float64(t_0 / fma(sin(a), Float64(fma(fma(-0.008333333333333333, Float64(b * b), 0.16666666666666666), Float64(b * b), -1.0) * b), Float64(cos(b) * cos(a))));
	else
		tmp = Float64(t_0 / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000000000.0], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.0], N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e11

   1. Initial program 53.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.7

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.7%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 56.7

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 56.7

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -1.05e11 < b < 7

   1. Initial program 98.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right)}, \cos b \cdot \cos a\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right) \cdot \color{blue}{b}, \cos b \cdot \cos a\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) - 1\right) \cdot \color{blue}{b}, \cos b \cdot \cos a\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \left({b}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot b, \cos b \cdot \cos a\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \left(\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) \cdot {b}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot b, \cos b \cdot \cos a\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \left(\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}\right) \cdot {b}^{2} + -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\frac{1}{6} + \frac{-1}{120} \cdot {b}^{2}, {b}^{2}, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\frac{-1}{120} \cdot {b}^{2} + \frac{1}{6}, {b}^{2}, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, {b}^{2}, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, b \cdot b, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, b \cdot b, \frac{1}{6}\right), {b}^{2}, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, b \cdot b, \frac{1}{6}\right), b \cdot b, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

      12. lift-*.f64 99.0

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, b \cdot b, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b, \cos b \cdot \cos a\right)} \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, b \cdot b, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b}, \cos b \cdot \cos a\right)} \]

   if 7 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\cos b, \cos a, \left(-t\_0\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.39)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 0.4)
       (/ (* r t_0) (fma (cos b) (cos a) (* (- t_0) (sin a))))
       (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.39) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 0.4) {
		tmp = (r * t_0) / fma(cos(b), cos(a), (-t_0 * sin(a)));
	} else {
		tmp = (r * sin(b)) / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.39)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / fma(cos(b), cos(a), Float64(Float64(-t_0) * sin(a))));
	else
		tmp = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.39], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[((-t$95$0) * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.39000000000000001

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 57.0

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 57.0

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 57.0

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\cos b \cdot \cos a} - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\cos b} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \color{blue}{\cos a} - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a}} \]

       6. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \color{blue}{\sin a}} \]

       7. cancel-sign-sub-inv N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)\right) \cdot \sin a}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)\right) \cdot \sin a\right)}} \]

       9. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)\right) \cdot \sin a\right)} \]

       10. lift-cos.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)\right) \cdot \sin a\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)\right) \cdot \sin a}\right)} \]

       12. lower-neg.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a\right)} \]

       13. lift-sin.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\cos b, \cos a, \left(-\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \color{blue}{\sin a}\right)} \]

   11. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a\right)}} \]

   if 0.40000000000000002 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\cos b \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.39)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 0.4)
       (/ (* r t_0) (- (* (cos b) (cos a)) (* t_0 (sin a))))
       (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.39) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((cos(b) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = (r * sin(b)) / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.39)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(cos(b) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.39], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.39000000000000001

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 57.0

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 57.0

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 57.0

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   if 0.40000000000000002 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ t_1 := \sin b \cdot r\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_1 \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right) \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\cos b - \sin b \cdot \sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b))
        (t_1 (* (sin b) r)))
   (if (<= b -0.39)
     (* t_1 (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 0.4)
       (/
        (* r t_0)
        (-
         (*
          (fma
           (fma
            (fma (* b b) -0.001388888888888889 0.041666666666666664)
            (* b b)
            -0.5)
           (* b b)
           1.0)
          (cos a))
         (* t_0 (sin a))))
       (/ t_1 (- (cos b) (* (sin b) (sin a))))))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double t_1 = sin(b) * r;
	double tmp;
	if (b <= -0.39) {
		tmp = t_1 * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((fma(fma(fma((b * b), -0.001388888888888889, 0.041666666666666664), (b * b), -0.5), (b * b), 1.0) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = t_1 / (cos(b) - (sin(b) * sin(a)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	t_1 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -0.39)
		tmp = Float64(t_1 * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(fma(fma(fma(Float64(b * b), -0.001388888888888889, 0.041666666666666664), Float64(b * b), -0.5), Float64(b * b), 1.0) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = Float64(t_1 / Float64(cos(b) - Float64(sin(b) * sin(a))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.39], N[(t$95$1 * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.39000000000000001

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 57.0

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 57.0

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 57.0

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       2. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left(\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) \cdot {b}^{2} + 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}, \color{blue}{{b}^{2}}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. sub-flip N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       6. metadata-eval N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \frac{-1}{2}, {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}, {b}^{2}, \frac{-1}{2}\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       8. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {b}^{2} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       9. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left({b}^{2} \cdot \frac{-1}{720} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       11. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       13. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       15. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       16. lift-*.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   if 0.40000000000000002 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.2

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      5. lift-sin.f64 56.2

         \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos b - \sin b \cdot \sin a} \]

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right) \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.39)
     (* (* (sin b) r) (/ 1.0 (- (cos b) (* (sin a) (sin b)))))
     (if (<= b 0.4)
       (/
        (* r t_0)
        (-
         (*
          (fma
           (fma
            (fma (* b b) -0.001388888888888889 0.041666666666666664)
            (* b b)
            -0.5)
           (* b b)
           1.0)
          (cos a))
         (* t_0 (sin a))))
       (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* (cos b) 1.0)))))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.39) {
		tmp = (sin(b) * r) * (1.0 / (cos(b) - (sin(a) * sin(b))));
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((fma(fma(fma((b * b), -0.001388888888888889, 0.041666666666666664), (b * b), -0.5), (b * b), 1.0) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = (r * sin(b)) / fma(sin(a), -sin(b), (cos(b) * 1.0));
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.39)
		tmp = Float64(Float64(sin(b) * r) * Float64(1.0 / Float64(cos(b) - Float64(sin(a) * sin(b)))));
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(fma(fma(fma(Float64(b * b), -0.001388888888888889, 0.041666666666666664), Float64(b * b), -0.5), Float64(b * b), 1.0) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(b) * 1.0)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.39], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] * N[(1.0 / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.39000000000000001

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 57.0

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin b \cdot r\right)} \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      8. lift-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin b} \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \sin a} \]

      9. lower-/.f64 57.0

         \[\leadsto \left(\sin b \cdot r\right) \cdot \color{blue}{\frac{1}{\cos b - \sin b \cdot \sin a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      16. lift-sin.f64 57.0

          \[\leadsto \left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\sin b \cdot r\right) \cdot \frac{1}{\cos b - \sin a \cdot \sin b}} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       2. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left(\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) \cdot {b}^{2} + 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}, \color{blue}{{b}^{2}}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. sub-flip N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       6. metadata-eval N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \frac{-1}{2}, {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}, {b}^{2}, \frac{-1}{2}\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       8. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {b}^{2} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       9. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left({b}^{2} \cdot \frac{-1}{720} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       11. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       13. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       15. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       16. lift-*.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   if 0.40000000000000002 < b

   1. Initial program 54.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right) \cdot \sin a}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right)} \cdot \sin a} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\color{blue}{\sin b}\right)\right) \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \color{blue}{\sin a}} \]

      8. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \cos b \cdot \cos a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \cos b \cdot \cos a} \]

      10. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\sin a \cdot \left(-1 \cdot \sin b\right)} + \cos b \cdot \cos a} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -1 \cdot \sin b, \cos b \cdot \cos a\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\sin a}, -1 \cdot \sin b, \cos b \cdot \cos a\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{\mathsf{neg}\left(\sin b\right)}, \cos b \cdot \cos a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \mathsf{neg}\left(\color{blue}{\sin b}\right), \cos b \cdot \cos a\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \color{blue}{-\sin b}, \cos b \cdot \cos a\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b} \cdot \cos a\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{\cos a}\right)} \]

      18. lift-*.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \color{blue}{\cos b \cdot \cos a}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \color{blue}{1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ t_1 := \frac{\sin b \cdot r}{\cos b - \sin b \cdot \sin a}\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right) \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b))
        (t_1 (/ (* (sin b) r) (- (cos b) (* (sin b) (sin a))))))
   (if (<= b -0.39)
     t_1
     (if (<= b 0.4)
       (/
        (* r t_0)
        (-
         (*
          (fma
           (fma
            (fma (* b b) -0.001388888888888889 0.041666666666666664)
            (* b b)
            -0.5)
           (* b b)
           1.0)
          (cos a))
         (* t_0 (sin a))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double t_1 = (sin(b) * r) / (cos(b) - (sin(b) * sin(a)));
	double tmp;
	if (b <= -0.39) {
		tmp = t_1;
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((fma(fma(fma((b * b), -0.001388888888888889, 0.041666666666666664), (b * b), -0.5), (b * b), 1.0) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	t_1 = Float64(Float64(sin(b) * r) / Float64(cos(b) - Float64(sin(b) * sin(a))))
	tmp = 0.0
	if (b <= -0.39)
		tmp = t_1;
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(fma(fma(fma(Float64(b * b), -0.001388888888888889, 0.041666666666666664), Float64(b * b), -0.5), Float64(b * b), 1.0) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.39], t$95$1, If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 0.40000000000000002 < b

   1. Initial program 54.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.6

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      5. lift-sin.f64 56.6

         \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos b - \sin b \cdot \sin a} \]

   8. Applied rewrites 56.6%

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       2. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left(\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}\right) \cdot {b}^{2} + 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) - \frac{1}{2}, \color{blue}{{b}^{2}}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. sub-flip N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       6. metadata-eval N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}\right) \cdot {b}^{2} + \frac{-1}{2}, {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {b}^{2}, {b}^{2}, \frac{-1}{2}\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       8. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {b}^{2} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       9. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left({b}^{2} \cdot \frac{-1}{720} + \frac{1}{24}, {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       11. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), {b}^{2}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       13. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       15. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{720}, \frac{1}{24}\right), b \cdot b, \frac{-1}{2}\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       16. lift-*.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.001388888888888889, 0.041666666666666664\right), b \cdot b, -0.5\right), b \cdot b, 1\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ t_1 := \frac{\sin b \cdot r}{\cos b - \sin b \cdot \sin a}\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot b, 1\right) \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b))
        (t_1 (/ (* (sin b) r) (- (cos b) (* (sin b) (sin a))))))
   (if (<= b -0.39)
     t_1
     (if (<= b 0.4)
       (/
        (* r t_0)
        (-
         (* (fma (fma (* b b) 0.041666666666666664 -0.5) (* b b) 1.0) (cos a))
         (* t_0 (sin a))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double t_1 = (sin(b) * r) / (cos(b) - (sin(b) * sin(a)));
	double tmp;
	if (b <= -0.39) {
		tmp = t_1;
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((fma(fma((b * b), 0.041666666666666664, -0.5), (b * b), 1.0) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	t_1 = Float64(Float64(sin(b) * r) / Float64(cos(b) - Float64(sin(b) * sin(a))))
	tmp = 0.0
	if (b <= -0.39)
		tmp = t_1;
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(fma(fma(Float64(b * b), 0.041666666666666664, -0.5), Float64(b * b), 1.0) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.39], t$95$1, If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 0.40000000000000002 < b

   1. Initial program 54.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.6

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

      5. lift-sin.f64 56.6

         \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos b - \sin b \cdot \sin a} \]

   8. Applied rewrites 56.6%

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b - \sin b \cdot \sin a} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\left(1 + {b}^{2} \cdot \left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right)\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left({b}^{2} \cdot \left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       2. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left(\left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right) \cdot {b}^{2} + 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}, \color{blue}{{b}^{2}}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. sub-flip N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       6. metadata-eval N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{24} + \frac{-1}{2}, {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{24}, \frac{-1}{2}\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       8. pow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       10. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       11. lift-*.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot b, 1\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\\ t_1 := r \cdot \frac{\sin b}{\cos b - \sin a \cdot \sin b}\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot b, 1\right) \cdot \cos a - t\_0 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           (* b b)
           1.0)
          b))
        (t_1 (* r (/ (sin b) (- (cos b) (* (sin a) (sin b)))))))
   (if (<= b -0.39)
     t_1
     (if (<= b 0.4)
       (/
        (* r t_0)
        (-
         (* (fma (fma (* b b) 0.041666666666666664 -0.5) (* b b) 1.0) (cos a))
         (* t_0 (sin a))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * b), 1.0) * b;
	double t_1 = r * (sin(b) / (cos(b) - (sin(a) * sin(b))));
	double tmp;
	if (b <= -0.39) {
		tmp = t_1;
	} else if (b <= 0.4) {
		tmp = (r * t_0) / ((fma(fma((b * b), 0.041666666666666664, -0.5), (b * b), 1.0) * cos(a)) - (t_0 * sin(a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * b), 1.0) * b)
	t_1 = Float64(r * Float64(sin(b) / Float64(cos(b) - Float64(sin(a) * sin(b)))))
	tmp = 0.0
	if (b <= -0.39)
		tmp = t_1;
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_0) / Float64(Float64(fma(fma(Float64(b * b), 0.041666666666666664, -0.5), Float64(b * b), 1.0) * cos(a)) - Float64(t_0 * sin(a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.39], t$95$1, If[LessEqual[b, 0.4], N[(N[(r * t$95$0), $MachinePrecision] / N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 0.40000000000000002 < b

   1. Initial program 54.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.2

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. lift-cos.f64 56.6

         \[\leadsto \frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a} \]

   6. Applied rewrites 56.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} - \sin b \cdot \sin a} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b - \sin b \cdot \sin a} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos b - \sin b \cdot \sin a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos b - \sin b \cdot \sin a}} \]

      6. lower-/.f64 N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b - \sin b \cdot \sin a}} \]

      7. lift-sin.f64 56.6

         \[\leadsto r \cdot \frac{\color{blue}{\sin b}}{\cos b - \sin b \cdot \sin a} \]

      8. lift-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b - \color{blue}{\sin b \cdot \sin a}} \]

      9. lift-sin.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b - \color{blue}{\sin b} \cdot \sin a} \]

      10. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b - \sin b \cdot \color{blue}{\sin a}} \]

      11. *-commutative N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      12. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b - \color{blue}{\sin a \cdot \sin b}} \]

      13. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b - \color{blue}{\sin a} \cdot \sin b} \]

      14. lift-sin.f64 56.6

          \[\leadsto r \cdot \frac{\sin b}{\cos b - \sin a \cdot \color{blue}{\sin b}} \]

   8. Applied rewrites 56.6%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos b - \sin a \cdot \sin b}} \]

   if -0.39000000000000001 < b < 0.40000000000000002

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      11. lower-sin.f64 99.8

          \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin a} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}\right) \cdot \sin a} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b\right) \cdot \sin a} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \cdot \sin a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      6. sub-flip N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      7. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      10. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \cdot \sin a} \]

      12. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\left(1 + {b}^{2} \cdot \left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right)\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left({b}^{2} \cdot \left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       2. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\left(\left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}\right) \cdot {b}^{2} + 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {b}^{2} - \frac{1}{2}, \color{blue}{{b}^{2}}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       4. sub-flip N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       5. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       6. metadata-eval N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{24} + \frac{-1}{2}, {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left({b}^{2}, \frac{1}{24}, \frac{-1}{2}\right), {\color{blue}{b}}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       8. pow2 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), {b}^{2}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       10. pow2 N/A

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{24}, \frac{-1}{2}\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

       11. lift-*.f64 99.6

           \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot \color{blue}{b}, 1\right) \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.041666666666666664, -0.5\right), b \cdot b, 1\right)} \cdot \cos a - \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.062:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076)
     t_0
     (if (<= b 0.062)
       (/ (* r (* (fma (* b b) -0.16666666666666666 1.0) b)) (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 0.062) {
		tmp = (r * (fma((b * b), -0.16666666666666666, 1.0) * b)) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 0.062)
		tmp = Float64(Float64(r * Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b)) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.076], t$95$0, If[LessEqual[b, 0.062], N[(N[(r * N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0759999999999999981 or 0.062 < b

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      3. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      4. lower-tan.f64 54.0

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{r \cdot \tan b} \]
   5. Step-by-step derivation

      1. *-commutative 54.0

         \[\leadsto r \cdot \tan b \]

      2. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      3. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      4. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      6. lift-tan.f64 54.0

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.0759999999999999981 < b < 0.062

   1. Initial program 98.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}\right)}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}\right)}{\cos \left(a + b\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left(\frac{-1}{6} \cdot {b}^{2} + 1\right) \cdot b\right)}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \frac{-1}{6} + 1\right) \cdot b\right)}{\cos \left(a + b\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left({b}^{2}, \frac{-1}{6}, 1\right) \cdot b\right)}{\cos \left(a + b\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right)}{\cos \left(a + b\right)} \]

      7. lower-*.f64 98.8

         \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}{\cos \left(a + b\right)} \]

   4. Applied rewrites 98.8%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.062:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076)
     t_0
     (if (<= b 0.062)
       (* (* (fma (* b b) -0.16666666666666666 1.0) b) (/ r (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 0.062) {
		tmp = (fma((b * b), -0.16666666666666666, 1.0) * b) * (r / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 0.062)
		tmp = Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) * Float64(r / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.076], t$95$0, If[LessEqual[b, 0.062], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0759999999999999981 or 0.062 < b

   1. Initial program 54.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      3. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      4. lower-tan.f64 54.0

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{r \cdot \tan b} \]
   5. Step-by-step derivation

      1. *-commutative 54.0

         \[\leadsto r \cdot \tan b \]

      2. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      3. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      4. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      6. lift-tan.f64 54.0

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.0759999999999999981 < b < 0.062

   1. Initial program 98.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos \left(a + b\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos \left(a + b\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

      5. lift-sin.f64 98.9

         \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos \left(a + b\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \cdot r}{\cos \left(a + b\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}\right) \cdot r}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}\right) \cdot r}{\cos \left(a + b\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot {b}^{2} + 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\left({b}^{2} \cdot \frac{-1}{6} + 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left({b}^{2}, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)} \]

      7. lower-*.f64 98.8

         \[\leadsto \frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)} \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)} \cdot r}{\cos \left(a + b\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}{\cos \left(a + b\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}}{\cos \left(a + b\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}{\cos \color{blue}{\left(a + b\right)}} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot r}{\color{blue}{\cos \left(a + b\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot \color{blue}{\frac{r}{\cos \left(a + b\right)}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b\right) \cdot \frac{r}{\color{blue}{\cos \left(a + b\right)}} \]

      9. lift-+.f64 98.8

         \[\leadsto \left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \color{blue}{\left(a + b\right)}} \]

   8. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076) t_0 (if (<= b 1.45e-6) (/ (* r b) (cos (+ a b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 1.45e-6) {
		tmp = (r * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-0.076d0)) then
        tmp = t_0
    else if (b <= 1.45d-6) then
        tmp = (r * b) / cos((a + b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 1.45e-6) {
		tmp = (r * b) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -0.076:
		tmp = t_0
	elif b <= 1.45e-6:
		tmp = (r * b) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 1.45e-6)
		tmp = Float64(Float64(r * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 1.45e-6)
		tmp = (r * b) / cos((a + b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.076], t$95$0, If[LessEqual[b, 1.45e-6], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0759999999999999981 or 1.4500000000000001e-6 < b

   1. Initial program 54.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      3. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      4. lower-tan.f64 54.1

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.1%

      \[\leadsto \color{blue}{r \cdot \tan b} \]
   5. Step-by-step derivation

      1. *-commutative 54.1

         \[\leadsto r \cdot \tan b \]

      2. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      3. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      4. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      6. lift-tan.f64 54.1

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.0759999999999999981 < b < 1.4500000000000001e-6

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.0%

      \[\leadsto \frac{r \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076) t_0 (if (<= b 1.45e-6) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 1.45e-6) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-0.076d0)) then
        tmp = t_0
    else if (b <= 1.45d-6) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -0.076) {
		tmp = t_0;
	} else if (b <= 1.45e-6) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -0.076:
		tmp = t_0
	elif b <= 1.45e-6:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 1.45e-6)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -0.076)
		tmp = t_0;
	elseif (b <= 1.45e-6)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.076], t$95$0, If[LessEqual[b, 1.45e-6], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0759999999999999981 or 1.4500000000000001e-6 < b

   1. Initial program 54.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

      3. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      4. lower-tan.f64 54.1

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.1%

      \[\leadsto \color{blue}{r \cdot \tan b} \]
   5. Step-by-step derivation

      1. *-commutative 54.1

         \[\leadsto r \cdot \tan b \]

      2. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      3. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      4. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      6. lift-tan.f64 54.1

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.0759999999999999981 < b < 1.4500000000000001e-6

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot \color{blue}{b} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{\sin a \cdot r}{0.5 + 0.5 \cdot \cos \left(2 \cdot a\right)}, \frac{r}{\cos a}\right) \cdot b} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos a} \cdot b \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{r}{\cos a} \cdot b \]

      2. lift-/.f64 98.9

         \[\leadsto \frac{r}{\cos a} \cdot b \]

   7. Applied rewrites 98.9%

      \[\leadsto \frac{r}{\cos a} \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* (sin b) r) (cos (+ a b))))

C

double code(double r, double a, double b) {
	return (sin(b) * r) / cos((a + b));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) * r) / cos((a + b))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) * r) / Math.cos((a + b));
}

Python

def code(r, a, b):
	return (math.sin(b) * r) / math.cos((a + b))

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / cos(Float64(a + b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (sin(b) * r) / cos((a + b));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos \left(a + b\right)} \]

   2. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos \left(a + b\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

   5. lift-sin.f64 77.0

      \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos \left(a + b\right)} \]

3. Applied rewrites 77.0%

   \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]
4. Add Preprocessing

Alternative 20: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin b \cdot \frac{r}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (sin b) (/ r (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return sin(b) * (r / cos((a + b)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sin(b) * (r / cos((a + b)))
end function

Java

public static double code(double r, double a, double b) {
	return Math.sin(b) * (r / Math.cos((a + b)));
}

Python

def code(r, a, b):
	return math.sin(b) * (r / math.cos((a + b)))

Julia

function code(r, a, b)
	return Float64(sin(b) * Float64(r / cos(Float64(a + b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = sin(b) * (r / cos((a + b)));
end

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   11. lower-sin.f64 99.5

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{r \cdot \sin b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   3. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \color{blue}{\sin b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   12. cos-sum-rev N/A

       \[\leadsto \frac{\sin b \cdot r}{\color{blue}{\cos \left(b + a\right)}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\sin b \cdot r}{\cos \color{blue}{\left(a + b\right)}} \]

   14. associate-/l* N/A

       \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(a + b\right)}} \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(a + b\right)}} \]

   16. lift-sin.f64 N/A

       \[\leadsto \color{blue}{\sin b} \cdot \frac{r}{\cos \left(a + b\right)} \]

   17. lower-/.f64 N/A

       \[\leadsto \sin b \cdot \color{blue}{\frac{r}{\cos \left(a + b\right)}} \]

   18. lift-cos.f64 N/A

       \[\leadsto \sin b \cdot \frac{r}{\color{blue}{\cos \left(a + b\right)}} \]

   19. lift-+.f64 76.9

       \[\leadsto \sin b \cdot \frac{r}{\cos \color{blue}{\left(a + b\right)}} \]

5. Applied rewrites 76.9%

   \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(a + b\right)}} \]
6. Add Preprocessing

Alternative 21: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan b \cdot r \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (tan b) r))

C

double code(double r, double a, double b) {
	return tan(b) * r;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = tan(b) * r
end function

Java

public static double code(double r, double a, double b) {
	return Math.tan(b) * r;
}

Python

def code(r, a, b):
	return math.tan(b) * r

Julia

function code(r, a, b)
	return Float64(tan(b) * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = tan(b) * r;
end

Wolfram

code[r_, a_, b_] := N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

   2. lower-*.f64 N/A

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]

   3. quot-tan N/A

      \[\leadsto r \cdot \tan b \]

   4. lower-tan.f64 60.1

      \[\leadsto r \cdot \tan b \]

4. Applied rewrites 60.1%

   \[\leadsto \color{blue}{r \cdot \tan b} \]
5. Step-by-step derivation

   1. *-commutative 60.1

      \[\leadsto r \cdot \tan b \]

   2. lift-*.f64 N/A

      \[\leadsto r \cdot \color{blue}{\tan b} \]

   3. lift-tan.f64 N/A

      \[\leadsto r \cdot \tan b \]

   4. *-commutative N/A

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   5. lower-*.f64 N/A

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   6. lift-tan.f64 60.1

      \[\leadsto \tan b \cdot r \]

6. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\tan b \cdot r} \]
7. Add Preprocessing

Alternative 22: 35.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ r \cdot b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r b))

C

double code(double r, double a, double b) {
	return r * b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * b
end function

Java

public static double code(double r, double a, double b) {
	return r * b;
}

Python

def code(r, a, b):
	return r * b

Julia

function code(r, a, b)
	return Float64(r * b)
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * b;
end

Wolfram

code[r_, a_, b_] := N[(r * b), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

2. Taylor expanded in b around 0

   \[\leadsto \color{blue}{b \cdot \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot \color{blue}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot \color{blue}{b} \]

4. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{\sin a \cdot r}{0.5 + 0.5 \cdot \cos \left(2 \cdot a\right)}, \frac{r}{\cos a}\right) \cdot b} \]

5. Taylor expanded in a around 0

   \[\leadsto r \cdot b \]
6. Step-by-step derivation

7. Applied rewrites 35.1%

   \[\leadsto r \cdot b \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))