rsin A (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 11.8s

Alternatives: 16

Speedup: 1.0×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= a -0.000102)
     (/ t_0 (cos a))
     (if (<= a 1.05e-5) (/ t_0 (cos b)) (* (/ (sin b) (cos a)) r)))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0 / cos(a);
	} else if (a <= 1.05e-5) {
		tmp = t_0 / cos(b);
	} else {
		tmp = (sin(b) / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (a <= (-0.000102d0)) then
        tmp = t_0 / cos(a)
    else if (a <= 1.05d-5) then
        tmp = t_0 / cos(b)
    else
        tmp = (sin(b) / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0 / Math.cos(a);
	} else if (a <= 1.05e-5) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if a <= -0.000102:
		tmp = t_0 / math.cos(a)
	elif a <= 1.05e-5:
		tmp = t_0 / math.cos(b)
	else:
		tmp = (math.sin(b) / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = Float64(t_0 / cos(a));
	elseif (a <= 1.05e-5)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = t_0 / cos(a);
	elseif (a <= 1.05e-5)
		tmp = t_0 / cos(b);
	else
		tmp = (sin(b) / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.000102], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-5], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999999e-4

   1. Initial program 51.4%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 51.7

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

   5. Applied rewrites 51.7%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.04999999999999994e-5 < a

   1. Initial program 60.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 47.6

         \[\leadsto \frac{r}{\frac{{\color{blue}{\left({\cos \left(a + b\right)}^{2}\right)}}^{0.5}}{\sin b}} \]

   6. Applied rewrites 47.6%

      \[\leadsto \frac{r}{\frac{\color{blue}{{\left({\cos \left(a + b\right)}^{2}\right)}^{0.5}}}{\sin b}} \]

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.4

         \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]

   9. Applied rewrites 47.4%

      \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 47.4

          \[\leadsto \color{blue}{\frac{\sin b}{{\left({\cos a}^{2}\right)}^{0.5}}} \cdot r \]

       8. lift-pow.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. unpow1 60.8

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

   11. Applied rewrites 60.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000102)
   (/ (* (sin b) r) (cos a))
   (if (<= a 1.05e-5) (* (/ r (cos b)) (sin b)) (* (/ (sin b) (cos a)) r))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (sin(b) * r) / cos(a);
	} else if (a <= 1.05e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (sin(b) / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.000102d0)) then
        tmp = (sin(b) * r) / cos(a)
    else if (a <= 1.05d-5) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (sin(b) / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	} else if (a <= 1.05e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000102:
		tmp = (math.sin(b) * r) / math.cos(a)
	elif a <= 1.05e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (math.sin(b) / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	elseif (a <= 1.05e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = (sin(b) * r) / cos(a);
	elseif (a <= 1.05e-5)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (sin(b) / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000102], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999999e-4

   1. Initial program 51.4%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 51.7

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

   5. Applied rewrites 51.7%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.04999999999999994e-5 < a

   1. Initial program 60.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 47.6

         \[\leadsto \frac{r}{\frac{{\color{blue}{\left({\cos \left(a + b\right)}^{2}\right)}}^{0.5}}{\sin b}} \]

   6. Applied rewrites 47.6%

      \[\leadsto \frac{r}{\frac{\color{blue}{{\left({\cos \left(a + b\right)}^{2}\right)}^{0.5}}}{\sin b}} \]

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.4

         \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]

   9. Applied rewrites 47.4%

      \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 47.4

          \[\leadsto \color{blue}{\frac{\sin b}{{\left({\cos a}^{2}\right)}^{0.5}}} \cdot r \]

       8. lift-pow.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. unpow1 60.8

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

   11. Applied rewrites 60.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000102)
   (* (/ r (cos a)) (sin b))
   (if (<= a 1.05e-5) (* (/ r (cos b)) (sin b)) (* (/ (sin b) (cos a)) r))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (r / cos(a)) * sin(b);
	} else if (a <= 1.05e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (sin(b) / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.000102d0)) then
        tmp = (r / cos(a)) * sin(b)
    else if (a <= 1.05d-5) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (sin(b) / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	} else if (a <= 1.05e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000102:
		tmp = (r / math.cos(a)) * math.sin(b)
	elif a <= 1.05e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (math.sin(b) / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	elseif (a <= 1.05e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = (r / cos(a)) * sin(b);
	elseif (a <= 1.05e-5)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (sin(b) / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000102], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999999e-4

   1. Initial program 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 30.8

         \[\leadsto \frac{r}{\frac{{\color{blue}{\left({\cos \left(a + b\right)}^{2}\right)}}^{0.5}}{\sin b}} \]

   6. Applied rewrites 30.8%

      \[\leadsto \frac{r}{\frac{\color{blue}{{\left({\cos \left(a + b\right)}^{2}\right)}^{0.5}}}{\sin b}} \]

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 30.8

         \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]

   9. Applied rewrites 30.8%

      \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 30.8

          \[\leadsto \color{blue}{\frac{r}{{\left({\cos a}^{2}\right)}^{0.5}}} \cdot \sin b \]

       6. lift-pow.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow-pow N/A

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

       9. metadata-eval N/A

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

       10. unpow1 51.6

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

   11. Applied rewrites 51.6%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.04999999999999994e-5 < a

   1. Initial program 60.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 47.6

         \[\leadsto \frac{r}{\frac{{\color{blue}{\left({\cos \left(a + b\right)}^{2}\right)}}^{0.5}}{\sin b}} \]

   6. Applied rewrites 47.6%

      \[\leadsto \frac{r}{\frac{\color{blue}{{\left({\cos \left(a + b\right)}^{2}\right)}^{0.5}}}{\sin b}} \]

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.4

         \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]

   9. Applied rewrites 47.4%

      \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 47.4

          \[\leadsto \color{blue}{\frac{\sin b}{{\left({\cos a}^{2}\right)}^{0.5}}} \cdot r \]

       8. lift-pow.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. unpow1 60.8

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

   11. Applied rewrites 60.8%

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

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos a)) (sin b))))
   (if (<= a -0.000102)
     t_0
     (if (<= a 1.05e-5) (* (/ r (cos b)) (sin b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(a)) * sin(b);
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0;
	} else if (a <= 1.05e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    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 / cos(a)) * sin(b)
    if (a <= (-0.000102d0)) then
        tmp = t_0
    else if (a <= 1.05d-5) then
        tmp = (r / cos(b)) * sin(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.cos(a)) * Math.sin(b);
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0;
	} else if (a <= 1.05e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(a)) * math.sin(b)
	tmp = 0
	if a <= -0.000102:
		tmp = t_0
	elif a <= 1.05e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(a)) * sin(b))
	tmp = 0.0
	if (a <= -0.000102)
		tmp = t_0;
	elseif (a <= 1.05e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(a)) * sin(b);
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = t_0;
	elseif (a <= 1.05e-5)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.000102], t$95$0, If[LessEqual[a, 1.05e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.01999999999999999e-4 or 1.04999999999999994e-5 < a

   1. Initial program 56.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 56.0

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

   4. Applied rewrites 56.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 39.2

         \[\leadsto \frac{r}{\frac{{\color{blue}{\left({\cos \left(a + b\right)}^{2}\right)}}^{0.5}}{\sin b}} \]

   6. Applied rewrites 39.2%

      \[\leadsto \frac{r}{\frac{\color{blue}{{\left({\cos \left(a + b\right)}^{2}\right)}^{0.5}}}{\sin b}} \]

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 39.1

         \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]

   9. Applied rewrites 39.1%

      \[\leadsto \frac{r}{\frac{{\left({\color{blue}{\cos a}}^{2}\right)}^{0.5}}{\sin b}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 39.1

          \[\leadsto \color{blue}{\frac{r}{{\left({\cos a}^{2}\right)}^{0.5}}} \cdot \sin b \]

       6. lift-pow.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow-pow N/A

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

       9. metadata-eval N/A

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

       10. unpow1 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -0.024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -0.024)
     t_0
     (if (<= b 3e-7)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -0.024) {
		tmp = t_0;
	} else if (b <= 3e-7) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -0.024)
		tmp = t_0;
	elseif (b <= 3e-7)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.024], t$95$0, If[LessEqual[b, 3e-7], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.024 or 2.9999999999999999e-7 < b

   1. Initial program 51.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -0.024 < b < 2.9999999999999999e-7

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

Alternative 7: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Final simplification 76.3%

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

Alternative 8: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 76.3

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

4. Applied rewrites 76.3%

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

Alternative 9: 77.0% 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

real(8) function code(r, a, b)
    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 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 76.3

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

4. Applied rewrites 76.3%

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

Alternative 10: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\left(-b\right) \cdot \sin a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 3.6e+16) (/ (* b r) (cos (+ a b))) (/ r (/ (* (- b) (sin a)) b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 3.6e+16) {
		tmp = (b * r) / cos((a + b));
	} else {
		tmp = r / ((-b * sin(a)) / b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.6d+16) then
        tmp = (b * r) / cos((a + b))
    else
        tmp = r / ((-b * sin(a)) / b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 3.6e+16) {
		tmp = (b * r) / Math.cos((a + b));
	} else {
		tmp = r / ((-b * Math.sin(a)) / b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 3.6e+16:
		tmp = (b * r) / math.cos((a + b))
	else:
		tmp = r / ((-b * math.sin(a)) / b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 3.6e+16)
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	else
		tmp = Float64(r / Float64(Float64(Float64(-b) * sin(a)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 3.6e+16)
		tmp = (b * r) / cos((a + b));
	else
		tmp = r / ((-b * sin(a)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 3.6e+16], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(N[((-b) * N[Sin[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.6e16

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 3.6e16 < b

   1. Initial program 53.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.3

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.0%

       \[\leadsto \frac{r}{\frac{\left(-\sin a\right) \cdot b}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\left(-b\right) \cdot \sin a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{-\sin a}\\ \mathbf{if}\;b \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (- (sin a)))))
   (if (<= b -7e+15) t_0 (if (<= b 5.2e+18) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / -sin(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    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(a)
    if (b <= (-7d+15)) then
        tmp = t_0
    else if (b <= 5.2d+18) 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(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / -math.sin(a)
	tmp = 0
	if b <= -7e+15:
		tmp = t_0
	elif b <= 5.2e+18:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(-sin(a)))
	tmp = 0.0
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		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(a);
	tmp = 0.0;
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = (b * r) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -7e+15], t$95$0, If[LessEqual[b, 5.2e+18], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7e15 or 5.2e18 < b

   1. Initial program 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.8

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.6

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.5%

       \[\leadsto \frac{r}{-\sin a} \]

   if -7e15 < b < 5.2e18

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 95.8

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.8%

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

Alternative 12: 53.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{-\sin a}\\ \mathbf{if}\;b \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (- (sin a)))))
   (if (<= b -7e+15) t_0 (if (<= b 5.2e+18) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / -sin(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    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(a)
    if (b <= (-7d+15)) then
        tmp = t_0
    else if (b <= 5.2d+18) 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(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / -math.sin(a)
	tmp = 0
	if b <= -7e+15:
		tmp = t_0
	elif b <= 5.2e+18:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(-sin(a)))
	tmp = 0.0
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		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(a);
	tmp = 0.0;
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -7e+15], t$95$0, If[LessEqual[b, 5.2e+18], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7e15 or 5.2e18 < b

   1. Initial program 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.8

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.6

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.5%

       \[\leadsto \frac{r}{-\sin a} \]

   if -7e15 < b < 5.2e18

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 13: 53.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{-\sin a}\\ \mathbf{if}\;b \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (- (sin a)))))
   (if (<= b -7e+15) t_0 (if (<= b 5.2e+18) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / -sin(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    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(a)
    if (b <= (-7d+15)) then
        tmp = t_0
    else if (b <= 5.2d+18) then
        tmp = (b / cos(a)) * r
    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(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / -math.sin(a)
	tmp = 0
	if b <= -7e+15:
		tmp = t_0
	elif b <= 5.2e+18:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(-sin(a)))
	tmp = 0.0
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r / -sin(a);
	tmp = 0.0;
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -7e+15], t$95$0, If[LessEqual[b, 5.2e+18], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7e15 or 5.2e18 < b

   1. Initial program 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.8

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.6

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.5%

       \[\leadsto \frac{r}{-\sin a} \]

   if -7e15 < b < 5.2e18

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 95.8

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.8%

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

Alternative 14: 37.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{-\sin a}\\ \mathbf{if}\;b \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;b \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (- (sin a)))))
   (if (<= b -7e+15) t_0 (if (<= b 5.2e+18) (* b r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / -sin(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = b * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    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(a)
    if (b <= (-7d+15)) then
        tmp = t_0
    else if (b <= 5.2d+18) then
        tmp = b * r
    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(a);
	double tmp;
	if (b <= -7e+15) {
		tmp = t_0;
	} else if (b <= 5.2e+18) {
		tmp = b * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / -math.sin(a)
	tmp = 0
	if b <= -7e+15:
		tmp = t_0
	elif b <= 5.2e+18:
		tmp = b * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(-sin(a)))
	tmp = 0.0
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = Float64(b * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r / -sin(a);
	tmp = 0.0;
	if (b <= -7e+15)
		tmp = t_0;
	elseif (b <= 5.2e+18)
		tmp = b * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -7e+15], t$95$0, If[LessEqual[b, 5.2e+18], N[(b * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7e15 or 5.2e18 < b

   1. Initial program 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.8

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.6

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.5%

       \[\leadsto \frac{r}{-\sin a} \]

   if -7e15 < b < 5.2e18

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.5%

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

Alternative 15: 52.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{-\sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 3.6e+16) (/ (* b r) (cos (+ a b))) (/ r (- (sin a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 3.6e+16) {
		tmp = (b * r) / cos((a + b));
	} else {
		tmp = r / -sin(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.6d+16) then
        tmp = (b * r) / cos((a + b))
    else
        tmp = r / -sin(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 3.6e+16) {
		tmp = (b * r) / Math.cos((a + b));
	} else {
		tmp = r / -Math.sin(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 3.6e+16:
		tmp = (b * r) / math.cos((a + b))
	else:
		tmp = r / -math.sin(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 3.6e+16)
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	else
		tmp = Float64(r / Float64(-sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 3.6e+16)
		tmp = (b * r) / cos((a + b));
	else
		tmp = r / -sin(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 3.6e+16], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.6e16

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 3.6e16 < b

   1. Initial program 53.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 10.3

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 10.0%

       \[\leadsto \frac{r}{-\sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 34.3% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))