rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 5.9s

Alternatives: 17

Speedup: 0.9×

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: 76.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan b}{\cos a} \cdot \frac{r}{1 - \tan a \cdot \tan b} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (/ (tan b) (cos a)) (/ r (- 1.0 (* (tan a) (tan b))))))

C

double code(double r, double a, double b) {
	return (tan(b) / cos(a)) * (r / (1.0 - (tan(a) * tan(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 = (tan(b) / cos(a)) * (r / (1.0d0 - (tan(a) * tan(b))))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.tan(b) / Math.cos(a)) * (r / (1.0 - (Math.tan(a) * Math.tan(b))));
}

Python

def code(r, a, b):
	return (math.tan(b) / math.cos(a)) * (r / (1.0 - (math.tan(a) * math.tan(b))))

Julia

function code(r, a, b)
	return Float64(Float64(tan(b) / cos(a)) * Float64(r / Float64(1.0 - Float64(tan(a) * tan(b)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (tan(b) / cos(a)) * (r / (1.0 - (tan(a) * tan(b))));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Tan[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - N[(N[Tan[a], $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\tan b}{\cos a} \cdot \frac{r}{1 - \tan a \cdot \tan b}} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\tan b \cdot r}{1 - \tan a \cdot \tan b}}{\cos a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (/ (* (tan b) r) (- 1.0 (* (tan a) (tan b)))) (cos a)))

C

double code(double r, double a, double b) {
	return ((tan(b) * r) / (1.0 - (tan(a) * tan(b)))) / cos(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 = ((tan(b) * r) / (1.0d0 - (tan(a) * tan(b)))) / cos(a)
end function

Java

public static double code(double r, double a, double b) {
	return ((Math.tan(b) * r) / (1.0 - (Math.tan(a) * Math.tan(b)))) / Math.cos(a);
}

Python

def code(r, a, b):
	return ((math.tan(b) * r) / (1.0 - (math.tan(a) * math.tan(b)))) / math.cos(a)

Julia

function code(r, a, b)
	return Float64(Float64(Float64(tan(b) * r) / Float64(1.0 - Float64(tan(a) * tan(b)))) / cos(a))
end

MATLAB

function tmp = code(r, a, b)
	tmp = ((tan(b) * r) / (1.0 - (tan(a) * tan(b)))) / cos(a);
end

Wolfram

code[r_, a_, b_] := N[(N[(N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - N[(N[Tan[a], $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\tan b \cdot r}{1 - \tan a \cdot \tan b}}{\cos a}} \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan b \cdot \left(-r\right)}{\cos a \cdot \mathsf{fma}\left(\tan a, \tan b, -1\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* (tan b) (- r)) (* (cos a) (fma (tan a) (tan b) -1.0))))

C

double code(double r, double a, double b) {
	return (tan(b) * -r) / (cos(a) * fma(tan(a), tan(b), -1.0));
}

Julia

function code(r, a, b)
	return Float64(Float64(tan(b) * Float64(-r)) / Float64(cos(a) * fma(tan(a), tan(b), -1.0)))
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Tan[b], $MachinePrecision] * (-r)), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[(N[Tan[a], $MachinePrecision] * N[Tan[b], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. frac-times N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. sub-flip N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\tan b \cdot \left(-r\right)}{\cos a \cdot \mathsf{fma}\left(\tan a, \tan b, -1\right)}} \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \tan b}{\left(1 - \tan a \cdot \tan b\right) \cdot \cos a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (tan b)) (* (- 1.0 (* (tan a) (tan b))) (cos a))))

C

double code(double r, double a, double b) {
	return (r * tan(b)) / ((1.0 - (tan(a) * tan(b))) * cos(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 * tan(b)) / ((1.0d0 - (tan(a) * tan(b))) * cos(a))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.tan(b)) / ((1.0 - (Math.tan(a) * Math.tan(b))) * Math.cos(a));
}

Python

def code(r, a, b):
	return (r * math.tan(b)) / ((1.0 - (math.tan(a) * math.tan(b))) * math.cos(a))

Julia

function code(r, a, b)
	return Float64(Float64(r * tan(b)) / Float64(Float64(1.0 - Float64(tan(a) * tan(b))) * cos(a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * tan(b)) / ((1.0 - (tan(a) * tan(b))) * cos(a));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(N[Tan[a], $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{r \cdot \tan b}{\left(1 - \tan a \cdot \tan b\right) \cdot \cos a}} \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \tan b \cdot \frac{r}{\left(1 - \tan a \cdot \tan b\right) \cdot \cos a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (tan b) (/ r (* (- 1.0 (* (tan a) (tan b))) (cos a)))))

C

double code(double r, double a, double b) {
	return tan(b) * (r / ((1.0 - (tan(a) * tan(b))) * cos(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 = tan(b) * (r / ((1.0d0 - (tan(a) * tan(b))) * cos(a)))
end function

Java

public static double code(double r, double a, double b) {
	return Math.tan(b) * (r / ((1.0 - (Math.tan(a) * Math.tan(b))) * Math.cos(a)));
}

Python

def code(r, a, b):
	return math.tan(b) * (r / ((1.0 - (math.tan(a) * math.tan(b))) * math.cos(a)))

Julia

function code(r, a, b)
	return Float64(tan(b) * Float64(r / Float64(Float64(1.0 - Float64(tan(a) * tan(b))) * cos(a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = tan(b) * (r / ((1.0 - (tan(a) * tan(b))) * cos(a)));
end

Wolfram

code[r_, a_, b_] := N[(N[Tan[b], $MachinePrecision] * N[(r / N[(N[(1.0 - N[(N[Tan[a], $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\tan b \cdot \frac{r}{\left(1 - \tan a \cdot \tan b\right) \cdot \cos a}} \]
8. Add Preprocessing

Alternative 6: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r}{1} \cdot \tan b}{\cos a} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* (/ r 1.0) (tan b)) (cos a)))

C

double code(double r, double a, double b) {
	return ((r / 1.0) * tan(b)) / cos(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 / 1.0d0) * tan(b)) / cos(a)
end function

Java

public static double code(double r, double a, double b) {
	return ((r / 1.0) * Math.tan(b)) / Math.cos(a);
}

Python

def code(r, a, b):
	return ((r / 1.0) * math.tan(b)) / math.cos(a)

Julia

function code(r, a, b)
	return Float64(Float64(Float64(r / 1.0) * tan(b)) / cos(a))
end

MATLAB

function tmp = code(r, a, b)
	tmp = ((r / 1.0) * tan(b)) / cos(a);
end

Wolfram

code[r_, a_, b_] := N[(N[(N[(r / 1.0), $MachinePrecision] * N[Tan[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 77.5%

   \[\leadsto \frac{\frac{r}{\color{blue}{1}} \cdot \tan b}{\cos a} \]
9. Add Preprocessing

Alternative 7: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 1200:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -3.1e-6)
   (* (/ (sin b) (cos a)) r)
   (if (<= a 1200.0) (/ (* r (sin b)) (cos b)) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.1e-6) {
		tmp = (sin(b) / cos(a)) * r;
	} else if (a <= 1200.0) {
		tmp = (r * sin(b)) / cos(b);
	} else {
		tmp = sin(b) * (r / cos(a));
	}
	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) :: tmp
    if (a <= (-3.1d-6)) then
        tmp = (sin(b) / cos(a)) * r
    else if (a <= 1200.0d0) then
        tmp = (r * sin(b)) / cos(b)
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.1e-6) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else if (a <= 1200.0) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -3.1e-6:
		tmp = (math.sin(b) / math.cos(a)) * r
	elif a <= 1200.0:
		tmp = (r * math.sin(b)) / math.cos(b)
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -3.1e-6)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	elseif (a <= 1200.0)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -3.1e-6)
		tmp = (sin(b) / cos(a)) * r;
	elseif (a <= 1200.0)
		tmp = (r * sin(b)) / cos(b);
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -3.1e-6], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[a, 1200.0], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.1e-6

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.8

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

   6. Applied rewrites 54.8%

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

   if -3.1e-6 < a < 1200

   1. Initial program 76.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 60.7%

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

   if 1200 < a

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.7

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

   6. Applied rewrites 54.7%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 76.6

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

3. Applied rewrites 76.6%

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

Alternative 9: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -560:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.8:\\ \;\;\;\;\frac{b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -560.0)
     t_0
     (if (<= b 5.8)
       (/ (* b (+ r (* -0.16666666666666666 (* (pow b 2.0) r)))) (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -560.0) {
		tmp = t_0;
	} else if (b <= 5.8) {
		tmp = (b * (r + (-0.16666666666666666 * (pow(b, 2.0) * 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 = (r * sin(b)) / 1.0d0
    if (b <= (-560.0d0)) then
        tmp = t_0
    else if (b <= 5.8d0) then
        tmp = (b * (r + ((-0.16666666666666666d0) * ((b ** 2.0d0) * 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 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -560.0) {
		tmp = t_0;
	} else if (b <= 5.8) {
		tmp = (b * (r + (-0.16666666666666666 * (Math.pow(b, 2.0) * r)))) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -560.0:
		tmp = t_0
	elif b <= 5.8:
		tmp = (b * (r + (-0.16666666666666666 * (math.pow(b, 2.0) * r)))) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -560.0)
		tmp = t_0;
	elseif (b <= 5.8)
		tmp = Float64(Float64(b * Float64(r + Float64(-0.16666666666666666 * Float64((b ^ 2.0) * r)))) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -560.0)
		tmp = t_0;
	elseif (b <= 5.8)
		tmp = (b * (r + (-0.16666666666666666 * ((b ^ 2.0) * r)))) / cos((a + b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -560.0], t$95$0, If[LessEqual[b, 5.8], N[(N[(b * N[(r + N[(-0.16666666666666666 * N[(N[Power[b, 2.0], $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -560 or 5.79999999999999982 < b

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -560 < b < 5.79999999999999982

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.5

         \[\leadsto \frac{b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)}{\cos \left(a + b\right)} \]

   4. Applied rewrites 50.5%

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

Alternative 10: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -560:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.8:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + -0.16666666666666666 \cdot {b}^{2}\right)\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -560.0)
     t_0
     (if (<= b 5.8)
       (/
        (* r (* b (+ 1.0 (* -0.16666666666666666 (pow b 2.0)))))
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -560.0) {
		tmp = t_0;
	} else if (b <= 5.8) {
		tmp = (r * (b * (1.0 + (-0.16666666666666666 * pow(b, 2.0))))) / 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 = (r * sin(b)) / 1.0d0
    if (b <= (-560.0d0)) then
        tmp = t_0
    else if (b <= 5.8d0) then
        tmp = (r * (b * (1.0d0 + ((-0.16666666666666666d0) * (b ** 2.0d0))))) / 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 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -560.0) {
		tmp = t_0;
	} else if (b <= 5.8) {
		tmp = (r * (b * (1.0 + (-0.16666666666666666 * Math.pow(b, 2.0))))) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -560.0:
		tmp = t_0
	elif b <= 5.8:
		tmp = (r * (b * (1.0 + (-0.16666666666666666 * math.pow(b, 2.0))))) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -560.0)
		tmp = t_0;
	elseif (b <= 5.8)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(-0.16666666666666666 * (b ^ 2.0))))) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -560.0)
		tmp = t_0;
	elseif (b <= 5.8)
		tmp = (r * (b * (1.0 + (-0.16666666666666666 * (b ^ 2.0))))) / cos((a + b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -560.0], t$95$0, If[LessEqual[b, 5.8], N[(N[(r * N[(b * N[(1.0 + N[(-0.16666666666666666 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -560 or 5.79999999999999982 < b

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -560 < b < 5.79999999999999982

   1. Initial program 76.6%

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.5

         \[\leadsto \frac{r \cdot \left(b \cdot \left(1 + -0.16666666666666666 \cdot {b}^{\color{blue}{2}}\right)\right)}{\cos \left(a + b\right)} \]

   4. Applied rewrites 50.5%

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

Alternative 11: 54.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -25000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.4:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -25000000000.0)
     t_0
     (if (<= b 3.4) (/ (* b r) (cos (+ a b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -25000000000.0) {
		tmp = t_0;
	} else if (b <= 3.4) {
		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 = (r * sin(b)) / 1.0d0
    if (b <= (-25000000000.0d0)) then
        tmp = t_0
    else if (b <= 3.4d0) 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 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -25000000000.0) {
		tmp = t_0;
	} else if (b <= 3.4) {
		tmp = (b * r) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -25000000000.0:
		tmp = t_0
	elif b <= 3.4:
		tmp = (b * r) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -25000000000.0)
		tmp = t_0;
	elseif (b <= 3.4)
		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 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -25000000000.0)
		tmp = t_0;
	elseif (b <= 3.4)
		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[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -25000000000.0], t$95$0, If[LessEqual[b, 3.4], 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 < -2.5e10 or 3.39999999999999991 < b

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -2.5e10 < b < 3.39999999999999991

   1. Initial program 76.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

Alternative 12: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -20000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -20000000000.0) t_0 (if (<= b 4.6) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -20000000000.0) {
		tmp = t_0;
	} else if (b <= 4.6) {
		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 = (r * sin(b)) / 1.0d0
    if (b <= (-20000000000.0d0)) then
        tmp = t_0
    else if (b <= 4.6d0) 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 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -20000000000.0) {
		tmp = t_0;
	} else if (b <= 4.6) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -20000000000.0:
		tmp = t_0
	elif b <= 4.6:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -20000000000.0)
		tmp = t_0;
	elseif (b <= 4.6)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -20000000000.0)
		tmp = t_0;
	elseif (b <= 4.6)
		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[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -20000000000.0], t$95$0, If[LessEqual[b, 4.6], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2e10 or 4.5999999999999996 < b

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -2e10 < b < 4.5999999999999996

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 50.8

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

   4. Applied rewrites 50.8%

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

Alternative 13: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -20000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -20000000000.0) t_0 (if (<= b 4.6) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -20000000000.0) {
		tmp = t_0;
	} else if (b <= 4.6) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * sin(b)) / 1.0d0
    if (b <= (-20000000000.0d0)) then
        tmp = t_0
    else if (b <= 4.6d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -20000000000.0) {
		tmp = t_0;
	} else if (b <= 4.6) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -20000000000.0:
		tmp = t_0
	elif b <= 4.6:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -20000000000.0)
		tmp = t_0;
	elseif (b <= 4.6)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -20000000000.0)
		tmp = t_0;
	elseif (b <= 4.6)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -20000000000.0], t$95$0, If[LessEqual[b, 4.6], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2e10 or 4.5999999999999996 < b

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -2e10 < b < 4.5999999999999996

   1. Initial program 76.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 50.8

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

   6. Applied rewrites 50.8%

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

Alternative 14: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in b around 0

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

   1. lower-cos.f64 54.7

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

4. Applied rewrites 54.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 54.7

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

6. Applied rewrites 54.7%

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

Alternative 15: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in b around 0

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

   1. lower-cos.f64 54.7

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

4. Applied rewrites 54.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 54.7

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

6. Applied rewrites 54.7%

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

Alternative 16: 39.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{1} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) 1.0))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / 1.0;
}

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)) / 1.0d0
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / 1.0;
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / 1.0

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / 1.0)
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / 1.0;
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in b around 0

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

   1. lower-cos.f64 54.7

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

4. Applied rewrites 54.7%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{r \cdot \sin b}{1} \]
6. Step-by-step derivation

7. Applied rewrites 39.1%

   \[\leadsto \frac{r \cdot \sin b}{1} \]
8. Add Preprocessing

Alternative 17: 35.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ r \cdot b \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 50.8

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

4. Applied rewrites 50.8%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{b \cdot r}{1 + \color{blue}{\frac{-1}{2} \cdot {a}^{2}}} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{b \cdot r}{1 + \frac{-1}{2} \cdot \color{blue}{{a}^{2}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{b \cdot r}{1 + \frac{-1}{2} \cdot {a}^{\color{blue}{2}}} \]

   3. lower-pow.f64 33.8

      \[\leadsto \frac{b \cdot r}{1 + -0.5 \cdot {a}^{2}} \]

7. Applied rewrites 33.8%

   \[\leadsto \frac{b \cdot r}{1 + \color{blue}{-0.5 \cdot {a}^{2}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{b \cdot r}{\color{blue}{1 + \frac{-1}{2} \cdot {a}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{b \cdot r}{\color{blue}{1} + \frac{-1}{2} \cdot {a}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{r \cdot b}{\color{blue}{1} + \frac{-1}{2} \cdot {a}^{2}} \]

   4. associate-/l* N/A

      \[\leadsto r \cdot \color{blue}{\frac{b}{1 + \frac{-1}{2} \cdot {a}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto r \cdot \color{blue}{\frac{b}{1 + \frac{-1}{2} \cdot {a}^{2}}} \]

   6. lower-/.f64 33.8

      \[\leadsto r \cdot \frac{b}{\color{blue}{1 + -0.5 \cdot {a}^{2}}} \]

   7. lift-+.f64 N/A

      \[\leadsto r \cdot \frac{b}{1 + \frac{-1}{2} \cdot \color{blue}{{a}^{2}}} \]

   8. +-commutative N/A

      \[\leadsto r \cdot \frac{b}{\frac{-1}{2} \cdot {a}^{2} + 1} \]

   9. lift-*.f64 N/A

      \[\leadsto r \cdot \frac{b}{\frac{-1}{2} \cdot {a}^{2} + 1} \]

   10. *-commutative N/A

       \[\leadsto r \cdot \frac{b}{{a}^{2} \cdot \frac{-1}{2} + 1} \]

   11. lower-fma.f64 33.8

       \[\leadsto r \cdot \frac{b}{\mathsf{fma}\left({a}^{2}, -0.5, 1\right)} \]

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 33.8

       \[\leadsto r \cdot \frac{b}{\mathsf{fma}\left(a \cdot a, -0.5, 1\right)} \]

9. Applied rewrites 33.8%

   \[\leadsto r \cdot \color{blue}{\frac{b}{\mathsf{fma}\left(a \cdot a, -0.5, 1\right)}} \]

10. Taylor expanded in a around 0

    \[\leadsto r \cdot b \]
11. Step-by-step derivation

12. Applied rewrites 35.0%

    \[\leadsto r \cdot b \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))