rsin A (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 1.1min

Alternatives: 5

Speedup: N/A×

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: 76.4% 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, N/A× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return (r * sin(b)) / ((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 = (r * sin(b)) / ((cos(b) * cos(a)) - (sin(b) * sin(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $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 78.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos a \cdot \cos b\\ t_1 := \sin a \cdot \sin b\\ \frac{r \cdot \mathsf{fma}\left(\sin b, t\_0, \sin b \cdot t\_1\right)}{{t\_0}^{2} - {t\_1}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (cos a) (cos b))) (t_1 (* (sin a) (sin b))))
   (/
    (* r (fma (sin b) t_0 (* (sin b) t_1)))
    (- (pow t_0 2.0) (pow t_1 2.0)))))

C

double code(double r, double a, double b) {
	double t_0 = cos(a) * cos(b);
	double t_1 = sin(a) * sin(b);
	return (r * fma(sin(b), t_0, (sin(b) * t_1))) / (pow(t_0, 2.0) - pow(t_1, 2.0));
}

Julia

function code(r, a, b)
	t_0 = Float64(cos(a) * cos(b))
	t_1 = Float64(sin(a) * sin(b))
	return Float64(Float64(r * fma(sin(b), t_0, Float64(sin(b) * t_1))) / Float64((t_0 ^ 2.0) - (t_1 ^ 2.0)))
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, N[(N[(r * N[(N[Sin[b], $MachinePrecision] * t$95$0 + N[(N[Sin[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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. cos-sum N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in r around 0

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

   1. pow-prod-down N/A

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

   2. pow2 N/A

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

   3. pow-prod-down N/A

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-diff-rev N/A

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

   6. distribute-lft-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 99.3

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

9. Applied rewrites 99.3%

   \[\leadsto \frac{r \cdot \mathsf{fma}\left(\sin b, \cos a \cdot \cos b, \sin b \cdot \left(\sin a \cdot \sin b\right)\right)}{{\left(\cos a \cdot \cos b\right)}^{\color{blue}{2}} - {\left(\sin a \cdot \sin b\right)}^{2}} \]
10. Add Preprocessing

Alternative 3: 76.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \left(\sin b \cdot \cos \left(a - b\right)\right)}{{\left(\cos a \cdot \cos b\right)}^{2} - {\left(\sin a \cdot \sin b\right)}^{2}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/
  (* r (* (sin b) (cos (- a b))))
  (- (pow (* (cos a) (cos b)) 2.0) (pow (* (sin a) (sin b)) 2.0))))

C

double code(double r, double a, double b) {
	return (r * (sin(b) * cos((a - b)))) / (pow((cos(a) * cos(b)), 2.0) - pow((sin(a) * sin(b)), 2.0));
}

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)))) / (((cos(a) * cos(b)) ** 2.0d0) - ((sin(a) * sin(b)) ** 2.0d0))
end function

Java

public static double code(double r, double a, double b) {
	return (r * (Math.sin(b) * Math.cos((a - b)))) / (Math.pow((Math.cos(a) * Math.cos(b)), 2.0) - Math.pow((Math.sin(a) * Math.sin(b)), 2.0));
}

Python

def code(r, a, b):
	return (r * (math.sin(b) * math.cos((a - b)))) / (math.pow((math.cos(a) * math.cos(b)), 2.0) - math.pow((math.sin(a) * math.sin(b)), 2.0))

Julia

function code(r, a, b)
	return Float64(Float64(r * Float64(sin(b) * cos(Float64(a - b)))) / Float64((Float64(cos(a) * cos(b)) ^ 2.0) - (Float64(sin(a) * sin(b)) ^ 2.0)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * (sin(b) * cos((a - b)))) / (((cos(a) * cos(b)) ^ 2.0) - ((sin(a) * sin(b)) ^ 2.0));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[(N[Sin[b], $MachinePrecision] * N[Cos[N[(a - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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. cos-sum N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in r around 0

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

   1. pow-prod-down N/A

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

   2. pow2 N/A

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

   3. pow-prod-down N/A

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\frac{r \cdot \left(\sin b \cdot \cos \left(a - b\right)\right)}{{\left(\cos a \cdot \cos b\right)}^{2} - {\left(\sin a \cdot \sin b\right)}^{2}}} \]
8. Add Preprocessing

Alternative 4: 74.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos a}\\ t_1 := \frac{\sin a \cdot r}{{\cos a}^{2}}\\ t_2 := r \cdot \tan b\\ t_3 := \mathsf{fma}\left(-0.5, t\_2, \left(-r\right) \cdot {\tan b}^{3}\right)\\ t_4 := r \cdot \left(\tan b \cdot \tan b\right)\\ t_5 := -0.16666666666666666 \cdot t\_0 - \mathsf{fma}\left(t\_0, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\\ \mathbf{if}\;b \leq -0.0018 \lor \neg \left(b \leq 0.25\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-a\right) \cdot \mathsf{fma}\left(t\_4, -0.5, \mathsf{fma}\left(0.16666666666666666, t\_4, \frac{t\_3 \cdot \sin b}{\cos b}\right)\right) - t\_3, a, t\_4\right), a, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(t\_1, -0.3333333333333333, \frac{t\_5 \cdot \sin a}{-\cos a}\right), t\_5\right), b, t\_1\right), b, t\_0\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (cos a)))
        (t_1 (/ (* (sin a) r) (pow (cos a) 2.0)))
        (t_2 (* r (tan b)))
        (t_3 (fma -0.5 t_2 (* (- r) (pow (tan b) 3.0))))
        (t_4 (* r (* (tan b) (tan b))))
        (t_5
         (-
          (* -0.16666666666666666 t_0)
          (fma t_0 -0.5 (/ (* (pow (sin a) 2.0) (- r)) (pow (cos a) 3.0))))))
   (if (or (<= b -0.0018) (not (<= b 0.25)))
     (fma
      (fma
       (-
        (*
         (- a)
         (fma
          t_4
          -0.5
          (fma 0.16666666666666666 t_4 (/ (* t_3 (sin b)) (cos b)))))
        t_3)
       a
       t_4)
      a
      t_2)
     (*
      (fma
       (fma
        (fma
         (- b)
         (fma t_1 -0.3333333333333333 (/ (* t_5 (sin a)) (- (cos a))))
         t_5)
        b
        t_1)
       b
       t_0)
      b))))

C

double code(double r, double a, double b) {
	double t_0 = r / cos(a);
	double t_1 = (sin(a) * r) / pow(cos(a), 2.0);
	double t_2 = r * tan(b);
	double t_3 = fma(-0.5, t_2, (-r * pow(tan(b), 3.0)));
	double t_4 = r * (tan(b) * tan(b));
	double t_5 = (-0.16666666666666666 * t_0) - fma(t_0, -0.5, ((pow(sin(a), 2.0) * -r) / pow(cos(a), 3.0)));
	double tmp;
	if ((b <= -0.0018) || !(b <= 0.25)) {
		tmp = fma(fma(((-a * fma(t_4, -0.5, fma(0.16666666666666666, t_4, ((t_3 * sin(b)) / cos(b))))) - t_3), a, t_4), a, t_2);
	} else {
		tmp = fma(fma(fma(-b, fma(t_1, -0.3333333333333333, ((t_5 * sin(a)) / -cos(a))), t_5), b, t_1), b, t_0) * b;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r / cos(a))
	t_1 = Float64(Float64(sin(a) * r) / (cos(a) ^ 2.0))
	t_2 = Float64(r * tan(b))
	t_3 = fma(-0.5, t_2, Float64(Float64(-r) * (tan(b) ^ 3.0)))
	t_4 = Float64(r * Float64(tan(b) * tan(b)))
	t_5 = Float64(Float64(-0.16666666666666666 * t_0) - fma(t_0, -0.5, Float64(Float64((sin(a) ^ 2.0) * Float64(-r)) / (cos(a) ^ 3.0))))
	tmp = 0.0
	if ((b <= -0.0018) || !(b <= 0.25))
		tmp = fma(fma(Float64(Float64(Float64(-a) * fma(t_4, -0.5, fma(0.16666666666666666, t_4, Float64(Float64(t_3 * sin(b)) / cos(b))))) - t_3), a, t_4), a, t_2);
	else
		tmp = Float64(fma(fma(fma(Float64(-b), fma(t_1, -0.3333333333333333, Float64(Float64(t_5 * sin(a)) / Float64(-cos(a)))), t_5), b, t_1), b, t_0) * b);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[a], $MachinePrecision] * r), $MachinePrecision] / N[Power[N[Cos[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * t$95$2 + N[((-r) * N[Power[N[Tan[b], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(r * N[(N[Tan[b], $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] - N[(t$95$0 * -0.5 + N[(N[(N[Power[N[Sin[a], $MachinePrecision], 2.0], $MachinePrecision] * (-r)), $MachinePrecision] / N[Power[N[Cos[a], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[b, -0.0018], N[Not[LessEqual[b, 0.25]], $MachinePrecision]], N[(N[(N[(N[((-a) * N[(t$95$4 * -0.5 + N[(0.16666666666666666 * t$95$4 + N[(N[(t$95$3 * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] * a + t$95$4), $MachinePrecision] * a + t$95$2), $MachinePrecision], N[(N[(N[(N[((-b) * N[(t$95$1 * -0.3333333333333333 + N[(N[(t$95$5 * N[Sin[a], $MachinePrecision]), $MachinePrecision] / (-N[Cos[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] * b + t$95$1), $MachinePrecision] * b + t$95$0), $MachinePrecision] * b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0018 or 0.25 < b

   1. Initial program 61.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-1}{2} \cdot \frac{r \cdot {\sin b}^{2}}{{\cos b}^{2}} + \left(\frac{1}{6} \cdot \frac{r \cdot {\sin b}^{2}}{{\cos b}^{2}} + \frac{\sin b \cdot \left(-1 \cdot \frac{r \cdot {\sin b}^{3}}{{\cos b}^{3}} + \frac{-1}{2} \cdot \frac{r \cdot \sin b}{\cos b}\right)}{\cos b}\right)\right)\right) - \left(-1 \cdot \frac{r \cdot {\sin b}^{3}}{{\cos b}^{3}} + \frac{-1}{2} \cdot \frac{r \cdot \sin b}{\cos b}\right)\right) - -1 \cdot \frac{r \cdot {\sin b}^{2}}{{\cos b}^{2}}\right) + \frac{r \cdot \sin b}{\cos b}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-a\right) \cdot \mathsf{fma}\left(r \cdot \left(\tan b \cdot \tan b\right), -0.5, \mathsf{fma}\left(0.16666666666666666, r \cdot \left(\tan b \cdot \tan b\right), \frac{\mathsf{fma}\left(-0.5, r \cdot \tan b, -r \cdot {\tan b}^{3}\right) \cdot \sin b}{\cos b}\right)\right) - \mathsf{fma}\left(-0.5, r \cdot \tan b, -r \cdot {\tan b}^{3}\right), a, 1 \cdot \left(r \cdot \left(\tan b \cdot \tan b\right)\right)\right), a, r \cdot \tan b\right)} \]

   if -0.0018 < b < 0.25

   1. Initial program 96.8%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{\sin a \cdot \left(\frac{-1}{6} \cdot \frac{r}{\cos a} - \left(-1 \cdot \frac{r \cdot {\sin a}^{2}}{{\cos a}^{3}} + \frac{-1}{2} \cdot \frac{r}{\cos a}\right)\right)}{\cos a} + \left(\frac{-1}{2} \cdot \frac{r \cdot \sin a}{{\cos a}^{2}} + \frac{1}{6} \cdot \frac{r \cdot \sin a}{{\cos a}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \frac{r}{\cos a}\right) - \left(-1 \cdot \frac{r \cdot {\sin a}^{2}}{{\cos a}^{3}} + \frac{-1}{2} \cdot \frac{r}{\cos a}\right)\right) - -1 \cdot \frac{r \cdot \sin a}{{\cos a}^{2}}\right) + \frac{r}{\cos a}\right)} \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(\frac{\sin a \cdot r}{{\cos a}^{2}}, -0.3333333333333333, -\frac{\left(-0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, -\frac{{\sin a}^{2} \cdot r}{{\cos a}^{3}}\right)\right) \cdot \sin a}{\cos a}\right), -0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, -\frac{{\sin a}^{2} \cdot r}{{\cos a}^{3}}\right)\right), b, 1 \cdot \frac{\sin a \cdot r}{{\cos a}^{2}}\right), b, \frac{r}{\cos a}\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0018 \lor \neg \left(b \leq 0.25\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-a\right) \cdot \mathsf{fma}\left(r \cdot \left(\tan b \cdot \tan b\right), -0.5, \mathsf{fma}\left(0.16666666666666666, r \cdot \left(\tan b \cdot \tan b\right), \frac{\mathsf{fma}\left(-0.5, r \cdot \tan b, \left(-r\right) \cdot {\tan b}^{3}\right) \cdot \sin b}{\cos b}\right)\right) - \mathsf{fma}\left(-0.5, r \cdot \tan b, \left(-r\right) \cdot {\tan b}^{3}\right), a, r \cdot \left(\tan b \cdot \tan b\right)\right), a, r \cdot \tan b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(\frac{\sin a \cdot r}{{\cos a}^{2}}, -0.3333333333333333, \frac{\left(-0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\right) \cdot \sin a}{-\cos a}\right), -0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\right), b, \frac{\sin a \cdot r}{{\cos a}^{2}}\right), b, \frac{r}{\cos a}\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos a}\\ t_1 := -0.16666666666666666 \cdot t\_0 - \mathsf{fma}\left(t\_0, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\\ t_2 := \frac{\sin a \cdot r}{{\cos a}^{2}}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(t\_2, -0.3333333333333333, \frac{t\_1 \cdot \sin a}{-\cos a}\right), t\_1\right), b, t\_2\right), b, t\_0\right) \cdot b \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (cos a)))
        (t_1
         (-
          (* -0.16666666666666666 t_0)
          (fma t_0 -0.5 (/ (* (pow (sin a) 2.0) (- r)) (pow (cos a) 3.0)))))
        (t_2 (/ (* (sin a) r) (pow (cos a) 2.0))))
   (*
    (fma
     (fma
      (fma
       (- b)
       (fma t_2 -0.3333333333333333 (/ (* t_1 (sin a)) (- (cos a))))
       t_1)
      b
      t_2)
     b
     t_0)
    b)))

C

double code(double r, double a, double b) {
	double t_0 = r / cos(a);
	double t_1 = (-0.16666666666666666 * t_0) - fma(t_0, -0.5, ((pow(sin(a), 2.0) * -r) / pow(cos(a), 3.0)));
	double t_2 = (sin(a) * r) / pow(cos(a), 2.0);
	return fma(fma(fma(-b, fma(t_2, -0.3333333333333333, ((t_1 * sin(a)) / -cos(a))), t_1), b, t_2), b, t_0) * b;
}

Julia

function code(r, a, b)
	t_0 = Float64(r / cos(a))
	t_1 = Float64(Float64(-0.16666666666666666 * t_0) - fma(t_0, -0.5, Float64(Float64((sin(a) ^ 2.0) * Float64(-r)) / (cos(a) ^ 3.0))))
	t_2 = Float64(Float64(sin(a) * r) / (cos(a) ^ 2.0))
	return Float64(fma(fma(fma(Float64(-b), fma(t_2, -0.3333333333333333, Float64(Float64(t_1 * sin(a)) / Float64(-cos(a)))), t_1), b, t_2), b, t_0) * b)
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] - N[(t$95$0 * -0.5 + N[(N[(N[Power[N[Sin[a], $MachinePrecision], 2.0], $MachinePrecision] * (-r)), $MachinePrecision] / N[Power[N[Cos[a], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[a], $MachinePrecision] * r), $MachinePrecision] / N[Power[N[Cos[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[((-b) * N[(t$95$2 * -0.3333333333333333 + N[(N[(t$95$1 * N[Sin[a], $MachinePrecision]), $MachinePrecision] / (-N[Cos[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * b + t$95$2), $MachinePrecision] * b + t$95$0), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{\sin a \cdot \left(\frac{-1}{6} \cdot \frac{r}{\cos a} - \left(-1 \cdot \frac{r \cdot {\sin a}^{2}}{{\cos a}^{3}} + \frac{-1}{2} \cdot \frac{r}{\cos a}\right)\right)}{\cos a} + \left(\frac{-1}{2} \cdot \frac{r \cdot \sin a}{{\cos a}^{2}} + \frac{1}{6} \cdot \frac{r \cdot \sin a}{{\cos a}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \frac{r}{\cos a}\right) - \left(-1 \cdot \frac{r \cdot {\sin a}^{2}}{{\cos a}^{3}} + \frac{-1}{2} \cdot \frac{r}{\cos a}\right)\right) - -1 \cdot \frac{r \cdot \sin a}{{\cos a}^{2}}\right) + \frac{r}{\cos a}\right)} \]

5. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(\frac{\sin a \cdot r}{{\cos a}^{2}}, -0.3333333333333333, -\frac{\left(-0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, -\frac{{\sin a}^{2} \cdot r}{{\cos a}^{3}}\right)\right) \cdot \sin a}{\cos a}\right), -0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, -\frac{{\sin a}^{2} \cdot r}{{\cos a}^{3}}\right)\right), b, 1 \cdot \frac{\sin a \cdot r}{{\cos a}^{2}}\right), b, \frac{r}{\cos a}\right) \cdot b} \]

6. Final simplification 48.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, \mathsf{fma}\left(\frac{\sin a \cdot r}{{\cos a}^{2}}, -0.3333333333333333, \frac{\left(-0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\right) \cdot \sin a}{-\cos a}\right), -0.16666666666666666 \cdot \frac{r}{\cos a} - \mathsf{fma}\left(\frac{r}{\cos a}, -0.5, \frac{{\sin a}^{2} \cdot \left(-r\right)}{{\cos a}^{3}}\right)\right), b, \frac{\sin a \cdot r}{{\cos a}^{2}}\right), b, \frac{r}{\cos a}\right) \cdot b \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))