rsin A (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 6.0s

Alternatives: 21

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

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(a) * cos(b))))
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[a], $MachinePrecision] * N[Cos[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. cos-neg-rev N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. cos-neg-rev N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. lift-cos.f64 99.5

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)}} \]
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. cos-neg-rev N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. cos-neg-rev N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. 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. fp-cancel-sub-sign-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. cos-neg-rev N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. cos-neg-rev N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. 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: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ \mathbf{if}\;b \leq -105000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\cos b, \cos a, \left(\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(b \cdot b\right) - 0.008333333333333333, b \cdot b, 0.16666666666666666\right) \cdot \left(b \cdot b\right) - 1\right) \cdot b\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b)))
        (t_1 (/ t_0 (fma (sin a) (- (sin b)) (* 1.0 (cos b))))))
   (if (<= b -105000000000.0)
     t_1
     (if (<= b 4.0)
       (/
        t_0
        (fma
         (cos b)
         (cos a)
         (*
          (*
           (-
            (*
             (fma
              (- (* 0.0001984126984126984 (* b b)) 0.008333333333333333)
              (* b b)
              0.16666666666666666)
             (* b b))
            1.0)
           b)
          (sin a))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double t_1 = t_0 / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double tmp;
	if (b <= -105000000000.0) {
		tmp = t_1;
	} else if (b <= 4.0) {
		tmp = t_0 / fma(cos(b), cos(a), ((((fma(((0.0001984126984126984 * (b * b)) - 0.008333333333333333), (b * b), 0.16666666666666666) * (b * b)) - 1.0) * b) * sin(a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	t_1 = Float64(t_0 / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	tmp = 0.0
	if (b <= -105000000000.0)
		tmp = t_1;
	elseif (b <= 4.0)
		tmp = Float64(t_0 / fma(cos(b), cos(a), Float64(Float64(Float64(Float64(fma(Float64(Float64(0.0001984126984126984 * Float64(b * b)) - 0.008333333333333333), Float64(b * b), 0.16666666666666666) * Float64(b * b)) - 1.0) * b) * sin(a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000000000.0], t$95$1, If[LessEqual[b, 4.0], N[(t$95$0 / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e11 or 4 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -1.05e11 < b < 4

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. 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 54.4%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(b \cdot b\right) - 0.008333333333333333, b \cdot b, 0.16666666666666666\right) \cdot \left(b \cdot b\right) - 1\right) \cdot b\right)} \cdot \sin a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.62:\\ \;\;\;\;\frac{r \cdot t\_1}{\cos b \cdot \cos a - t\_1 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* 1.0 (cos b)))))
        (t_1
         (*
          (fma
           (-
            (*
             (fma (* b b) -0.0001984126984126984 0.008333333333333333)
             (* b b))
            0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.5)
     t_0
     (if (<= b 0.62)
       (/ (* r t_1) (- (* (cos b) (cos a)) (* t_1 (sin a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double t_1 = fma(((fma((b * b), -0.0001984126984126984, 0.008333333333333333) * (b * b)) - 0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.5) {
		tmp = t_0;
	} else if (b <= 0.62) {
		tmp = (r * t_1) / ((cos(b) * cos(a)) - (t_1 * sin(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	t_1 = Float64(fma(Float64(Float64(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 = t_0;
	elseif (b <= 0.62)
		tmp = Float64(Float64(r * t_1) / Float64(Float64(cos(b) * cos(a)) - Float64(t_1 * sin(a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.5], t$95$0, If[LessEqual[b, 0.62], N[(N[(r * t$95$1), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.5 or 0.619999999999999996 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -0.5 < b < 0.619999999999999996

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. 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 52.1%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 51.5%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ \mathbf{if}\;b \leq -105000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\cos b, \cos a, \left(\left(\mathsf{fma}\left(-0.008333333333333333, b \cdot b, 0.16666666666666666\right) \cdot \left(b \cdot b\right) - 1\right) \cdot b\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b)))
        (t_1 (/ t_0 (fma (sin a) (- (sin b)) (* 1.0 (cos b))))))
   (if (<= b -105000000000.0)
     t_1
     (if (<= b 7.0)
       (/
        t_0
        (fma
         (cos b)
         (cos a)
         (*
          (*
           (-
            (* (fma -0.008333333333333333 (* b b) 0.16666666666666666) (* b b))
            1.0)
           b)
          (sin a))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double t_1 = t_0 / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double tmp;
	if (b <= -105000000000.0) {
		tmp = t_1;
	} else if (b <= 7.0) {
		tmp = t_0 / fma(cos(b), cos(a), ((((fma(-0.008333333333333333, (b * b), 0.16666666666666666) * (b * b)) - 1.0) * b) * sin(a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	t_1 = Float64(t_0 / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	tmp = 0.0
	if (b <= -105000000000.0)
		tmp = t_1;
	elseif (b <= 7.0)
		tmp = Float64(t_0 / fma(cos(b), cos(a), Float64(Float64(Float64(Float64(fma(-0.008333333333333333, Float64(b * b), 0.16666666666666666) * Float64(b * b)) - 1.0) * b) * sin(a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000000000.0], t$95$1, If[LessEqual[b, 7.0], N[(t$95$0 / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[(N[(N[(N[(-0.008333333333333333 * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e11 or 7 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -1.05e11 < b < 7

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. 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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 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. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 54.9

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

   8. Applied rewrites 54.9%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\left(\mathsf{fma}\left(-0.008333333333333333, b \cdot b, 0.16666666666666666\right) \cdot \left(b \cdot b\right) - 1\right) \cdot b\right)} \cdot \sin a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ t_1 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_1}{\mathsf{fma}\left(\cos b, \cos a, \left(-t\_1\right) \cdot \sin a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* 1.0 (cos b)))))
        (t_1
         (*
          (fma
           (- (* 0.008333333333333333 (* b b)) 0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.39)
     t_0
     (if (<= b 0.4)
       (/ (* r t_1) (fma (cos b) (cos a) (* (- t_1) (sin a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double t_1 = fma(((0.008333333333333333 * (b * b)) - 0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.39) {
		tmp = t_0;
	} else if (b <= 0.4) {
		tmp = (r * t_1) / fma(cos(b), cos(a), (-t_1 * sin(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	t_1 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(b * b)) - 0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.39)
		tmp = t_0;
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_1) / fma(cos(b), cos(a), Float64(Float64(-t_1) * sin(a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.39], t$95$0, If[LessEqual[b, 0.4], N[(N[(r * t$95$1), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[((-t$95$1) * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 0.40000000000000002 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -0.39000000000000001 < b < 0.40000000000000002

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 51.7

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

   11. Applied rewrites 51.7%

       \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}\right) \cdot \sin a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ t_1 := \mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.39:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.4:\\ \;\;\;\;\frac{r \cdot t\_1}{\cos b \cdot \cos a - t\_1 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* 1.0 (cos b)))))
        (t_1
         (*
          (fma
           (- (* (* b b) 0.008333333333333333) 0.16666666666666666)
           (* b b)
           1.0)
          b)))
   (if (<= b -0.39)
     t_0
     (if (<= b 0.4)
       (/ (* r t_1) (- (* (cos b) (cos a)) (* t_1 (sin a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double t_1 = fma((((b * b) * 0.008333333333333333) - 0.16666666666666666), (b * b), 1.0) * b;
	double tmp;
	if (b <= -0.39) {
		tmp = t_0;
	} else if (b <= 0.4) {
		tmp = (r * t_1) / ((cos(b) * cos(a)) - (t_1 * sin(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	t_1 = Float64(fma(Float64(Float64(Float64(b * b) * 0.008333333333333333) - 0.16666666666666666), Float64(b * b), 1.0) * b)
	tmp = 0.0
	if (b <= -0.39)
		tmp = t_0;
	elseif (b <= 0.4)
		tmp = Float64(Float64(r * t_1) / Float64(Float64(cos(b) * cos(a)) - Float64(t_1 * sin(a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.39], t$95$0, If[LessEqual[b, 0.4], N[(N[(r * t$95$1), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 0.40000000000000002 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -0.39000000000000001 < b < 0.40000000000000002

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. 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. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 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. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.1

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.7

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

   9. Applied rewrites 51.7%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)} \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, 1 \cdot \cos b\right)}\\ t_1 := \mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.28:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.25:\\ \;\;\;\;\frac{r \cdot t\_1}{\cos b \cdot \cos a - t\_1 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (fma (sin a) (- (sin b)) (* 1.0 (cos b)))))
        (t_1 (* (fma (* b b) -0.16666666666666666 1.0) b)))
   (if (<= b -0.28)
     t_0
     (if (<= b 0.25)
       (/ (* r t_1) (- (* (cos b) (cos a)) (* t_1 (sin a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / fma(sin(a), -sin(b), (1.0 * cos(b)));
	double t_1 = fma((b * b), -0.16666666666666666, 1.0) * b;
	double tmp;
	if (b <= -0.28) {
		tmp = t_0;
	} else if (b <= 0.25) {
		tmp = (r * t_1) / ((cos(b) * cos(a)) - (t_1 * sin(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(1.0 * cos(b))))
	t_1 = Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b)
	tmp = 0.0
	if (b <= -0.28)
		tmp = t_0;
	elseif (b <= 0.25)
		tmp = Float64(Float64(r * t_1) / Float64(Float64(cos(b) * cos(a)) - Float64(t_1 * sin(a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(1.0 * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.28], t$95$0, If[LessEqual[b, 0.25], N[(N[(r * t$95$1), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.28000000000000003 or 0.25 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. lift-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.3%

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

   if -0.28000000000000003 < b < 0.25

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.0

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

   6. Applied rewrites 52.0%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.1

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

   9. Applied rewrites 52.1%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)} \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\cos b \cdot 1 - \sin b \cdot \sin a}\\ t_1 := \mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\\ \mathbf{if}\;b \leq -0.28:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.25:\\ \;\;\;\;\frac{r \cdot t\_1}{\cos b \cdot \cos a - t\_1 \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (- (* (cos b) 1.0) (* (sin b) (sin a)))))
        (t_1 (* (fma (* b b) -0.16666666666666666 1.0) b)))
   (if (<= b -0.28)
     t_0
     (if (<= b 0.25)
       (/ (* r t_1) (- (* (cos b) (cos a)) (* t_1 (sin a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / ((cos(b) * 1.0) - (sin(b) * sin(a)));
	double t_1 = fma((b * b), -0.16666666666666666, 1.0) * b;
	double tmp;
	if (b <= -0.28) {
		tmp = t_0;
	} else if (b <= 0.25) {
		tmp = (r * t_1) / ((cos(b) * cos(a)) - (t_1 * sin(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / Float64(Float64(cos(b) * 1.0) - Float64(sin(b) * sin(a))))
	t_1 = Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b)
	tmp = 0.0
	if (b <= -0.28)
		tmp = t_0;
	elseif (b <= 0.25)
		tmp = Float64(Float64(r * t_1) / Float64(Float64(cos(b) * cos(a)) - Float64(t_1 * sin(a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -0.28], t$95$0, If[LessEqual[b, 0.25], N[(N[(r * t$95$1), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.28000000000000003 or 0.25 < b

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \color{blue}{1} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 61.3%

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

   if -0.28000000000000003 < b < 0.25

   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. cos-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. cos-neg-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. 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. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.0

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

   6. Applied rewrites 52.0%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.1

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

   9. Applied rewrites 52.1%

      \[\leadsto \frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)} \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 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 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. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      2. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      3. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lift-tan.f64 60.1

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 60.1%

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   if -0.0759999999999999981 < b < 0.062

   1. Initial program 77.0%

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

   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 51.6

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

   4. Applied rewrites 51.6%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076) t_0 (if (<= b 1.45e-6) (/ (* 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 <= 1.45e-6) {
		tmp = (b * 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 = (b * 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 = (b * 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 = (b * 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(b * r) / 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 = (b * 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[(b * r), $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 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. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      2. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      3. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lift-tan.f64 60.1

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 60.1%

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   if -0.0759999999999999981 < b < 1.4500000000000001e-6

   1. Initial program 77.0%

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

   2. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot r}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 51.9

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

   4. Applied rewrites 51.9%

      \[\leadsto \frac{\color{blue}{b \cdot r}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.076) t_0 (if (<= b 1.45e-6) (* b (/ r (cos a))) 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 = b * (r / cos(a));
	} 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 = b * (r / cos(a))
    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 = b * (r / Math.cos(a));
	} 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 = b * (r / math.cos(a))
	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(b * Float64(r / cos(a)));
	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 = b * (r / cos(a));
	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[(b * N[(r / N[Cos[a], $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 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. lift-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      2. lift-tan.f64 N/A

         \[\leadsto r \cdot \tan b \]

      3. *-commutative N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan b \cdot \color{blue}{r} \]

      5. lift-tan.f64 60.1

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 60.1%

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   if -0.0759999999999999981 < b < 1.4500000000000001e-6

   1. Initial program 77.0%

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

   2. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 51.9

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

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 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 15: 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. cos-neg-rev N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. cos-neg-rev N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. 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. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 N/A

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

   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 16: 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. lift-*.f64 N/A

      \[\leadsto r \cdot \color{blue}{\tan b} \]

   2. lift-tan.f64 N/A

      \[\leadsto r \cdot \tan b \]

   3. *-commutative N/A

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   4. lower-*.f64 N/A

      \[\leadsto \tan b \cdot \color{blue}{r} \]

   5. lift-tan.f64 60.1

      \[\leadsto \tan b \cdot r \]

6. Applied rewrites 60.1%

   \[\leadsto \tan b \cdot \color{blue}{r} \]
7. Add Preprocessing

Alternative 17: 34.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(r, b, \left(\left(\left(b \cdot b\right) \cdot r\right) \cdot 0.3333333333333333\right) \cdot b\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (fma r b (* (* (* (* b b) r) 0.3333333333333333) b)))

C

double code(double r, double a, double b) {
	return fma(r, b, ((((b * b) * r) * 0.3333333333333333) * b));
}

Julia

function code(r, a, b)
	return fma(r, b, Float64(Float64(Float64(Float64(b * b) * r) * 0.3333333333333333) * b))
end

Wolfram

code[r_, a_, b_] := N[(r * b + N[(N[(N[(N[(b * b), $MachinePrecision] * r), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * b), $MachinePrecision]), $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. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{\left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   7. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

7. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot \color{blue}{b} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   4. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   6. +-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   7. *-commutative N/A

      \[\leadsto b \cdot \left(r + \color{blue}{\frac{1}{3} \cdot \left({b}^{2} \cdot r\right)}\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto r \cdot b + \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot \color{blue}{b} \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(r, b, \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(r, b, \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(r, b, \left(\left({b}^{2} \cdot r\right) \cdot \frac{1}{3}\right) \cdot b\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(r, b, \left(\left({b}^{2} \cdot r\right) \cdot \frac{1}{3}\right) \cdot b\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{fma}\left(r, b, \left(\left(\left(b \cdot b\right) \cdot r\right) \cdot \frac{1}{3}\right) \cdot b\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(r, b, \left(\left(\left(b \cdot b\right) \cdot r\right) \cdot \frac{1}{3}\right) \cdot b\right) \]

   15. lift-*.f64 34.8

       \[\leadsto \mathsf{fma}\left(r, b, \left(\left(\left(b \cdot b\right) \cdot r\right) \cdot 0.3333333333333333\right) \cdot b\right) \]

9. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(r, b, \left(\left(\left(b \cdot b\right) \cdot r\right) \cdot 0.3333333333333333\right) \cdot b\right) \]
10. Add Preprocessing

Alternative 18: 34.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, b \cdot \left(b \cdot r\right), r\right) \cdot b \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (fma 0.3333333333333333 (* b (* b r)) r) b))

C

double code(double r, double a, double b) {
	return fma(0.3333333333333333, (b * (b * r)), r) * b;
}

Julia

function code(r, a, b)
	return Float64(fma(0.3333333333333333, Float64(b * Float64(b * r)), r) * b)
end

Wolfram

code[r_, a_, b_] := N[(N[(0.3333333333333333 * N[(b * N[(b * r), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision] * b), $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. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{\left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   7. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

7. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot \color{blue}{b} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, b \cdot \left(b \cdot r\right), r\right) \cdot b \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, b \cdot \left(b \cdot r\right), r\right) \cdot b \]

   5. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, b \cdot \left(b \cdot r\right), r\right) \cdot b \]

9. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, b \cdot \left(b \cdot r\right), r\right) \cdot b \]
10. Add Preprocessing

Alternative 19: 34.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(b \cdot b, 0.3333333333333333, 1\right) \cdot r\right) \cdot b \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (* (fma (* b b) 0.3333333333333333 1.0) r) b))

C

double code(double r, double a, double b) {
	return (fma((b * b), 0.3333333333333333, 1.0) * r) * b;
}

Julia

function code(r, a, b)
	return Float64(Float64(fma(Float64(b * b), 0.3333333333333333, 1.0) * r) * b)
end

Wolfram

code[r_, a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * r), $MachinePrecision] * b), $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. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{\left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   7. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

7. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot \color{blue}{b} \]

8. Taylor expanded in r around 0

   \[\leadsto \left(r \cdot \left(1 + \frac{1}{3} \cdot {b}^{2}\right)\right) \cdot b \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot r\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot r\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot {b}^{2} + 1\right) \cdot r\right) \cdot b \]

   4. *-commutative N/A

      \[\leadsto \left(\left({b}^{2} \cdot \frac{1}{3} + 1\right) \cdot r\right) \cdot b \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left({b}^{2}, \frac{1}{3}, 1\right) \cdot r\right) \cdot b \]

   6. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(b \cdot b, \frac{1}{3}, 1\right) \cdot r\right) \cdot b \]

   7. lift-*.f64 34.8

      \[\leadsto \left(\mathsf{fma}\left(b \cdot b, 0.3333333333333333, 1\right) \cdot r\right) \cdot b \]

10. Applied rewrites 34.8%

    \[\leadsto \left(\mathsf{fma}\left(b \cdot b, 0.3333333333333333, 1\right) \cdot r\right) \cdot b \]
11. Add Preprocessing

Alternative 20: 15.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(b \cdot b\right) \cdot \left(b \cdot r\right)\right) \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (* (* b b) (* b r)) 0.3333333333333333))

C

double code(double r, double a, double b) {
	return ((b * b) * (b * r)) * 0.3333333333333333;
}

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 = ((b * b) * (b * r)) * 0.3333333333333333d0
end function

Java

public static double code(double r, double a, double b) {
	return ((b * b) * (b * r)) * 0.3333333333333333;
}

Python

def code(r, a, b):
	return ((b * b) * (b * r)) * 0.3333333333333333

Julia

function code(r, a, b)
	return Float64(Float64(Float64(b * b) * Float64(b * r)) * 0.3333333333333333)
end

MATLAB

function tmp = code(r, a, b)
	tmp = ((b * b) * (b * r)) * 0.3333333333333333;
end

Wolfram

code[r_, a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * N[(b * r), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{\left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   7. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

7. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot \color{blue}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{1}{3} \cdot \left({b}^{3} \cdot \color{blue}{r}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   3. lower-*.f64 N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   4. unpow3 N/A

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   5. pow2 N/A

      \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   7. pow2 N/A

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   8. lift-*.f64 15.7

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot 0.3333333333333333 \]

10. Applied rewrites 15.7%

    \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot 0.3333333333333333 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    3. pow2 N/A

       \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    5. associate-*l* N/A

       \[\leadsto \left({b}^{2} \cdot \left(b \cdot r\right)\right) \cdot \frac{1}{3} \]

    6. lower-*.f64 N/A

       \[\leadsto \left({b}^{2} \cdot \left(b \cdot r\right)\right) \cdot \frac{1}{3} \]

    7. pow2 N/A

       \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot r\right)\right) \cdot \frac{1}{3} \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot r\right)\right) \cdot \frac{1}{3} \]

    9. lift-*.f64 15.7

       \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot r\right)\right) \cdot 0.3333333333333333 \]

12. Applied rewrites 15.7%

    \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot r\right)\right) \cdot 0.3333333333333333 \]
13. Add Preprocessing

Alternative 21: 15.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(0.3333333333333333 \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (* 0.3333333333333333 (* (* b b) b)) r))

C

double code(double r, double a, double b) {
	return (0.3333333333333333 * ((b * b) * 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 = (0.3333333333333333d0 * ((b * b) * b)) * r
end function

Java

public static double code(double r, double a, double b) {
	return (0.3333333333333333 * ((b * b) * b)) * r;
}

Python

def code(r, a, b):
	return (0.3333333333333333 * ((b * b) * b)) * r

Julia

function code(r, a, b)
	return Float64(Float64(0.3333333333333333 * Float64(Float64(b * b) * b)) * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = (0.3333333333333333 * ((b * b) * b)) * r;
end

Wolfram

code[r_, a_, b_] := N[(N[(0.3333333333333333 * N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $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. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{\left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(r + \frac{1}{3} \cdot \left({b}^{2} \cdot r\right)\right) \cdot b \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \left({b}^{2} \cdot r\right) + r\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, {b}^{2} \cdot r, r\right) \cdot b \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

   7. lower-*.f64 34.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot b \]

7. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \left(b \cdot b\right) \cdot r, r\right) \cdot \color{blue}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{1}{3} \cdot \left({b}^{3} \cdot \color{blue}{r}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   3. lower-*.f64 N/A

      \[\leadsto \left({b}^{3} \cdot r\right) \cdot \frac{1}{3} \]

   4. unpow3 N/A

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   5. pow2 N/A

      \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   7. pow2 N/A

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

   8. lift-*.f64 15.7

      \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot 0.3333333333333333 \]

10. Applied rewrites 15.7%

    \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot 0.3333333333333333 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \cdot \frac{1}{3} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot r\right) \]

    6. associate-*r* N/A

       \[\leadsto \left(\frac{1}{3} \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{3} \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{3} \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{3} \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]

    10. lift-*.f64 15.7

        \[\leadsto \left(0.3333333333333333 \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]

12. Applied rewrites 15.7%

    \[\leadsto \left(0.3333333333333333 \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot r \]
13. 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))))