rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 13.0s

Alternatives: 14

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: 76.2% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos b) (cos a) (* (sin b) (- 0.0 (sin a))))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(b), cos(a), (sin(b) * (0.0 - sin(a))));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), Float64(sin(b) * Float64(0.0 - sin(a)))))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[(0.0 - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. fmm-def N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \cos \color{blue}{a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\sin b, \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(0 - \sin a\right)\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \sin a\right)\right)\right)\right) \]

   11. sin-lowering-sin.f64 99.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   2. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\sin b, \sin a\right)\right)\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right)\right)\right) \]

   6. sin-lowering-sin.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

   8. sin-lowering-sin.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

6. Applied egg-rr 75.6%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\cos b \cdot \cos a\right), \left(\sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos b, \cos a\right), \left(\sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right), \left(\sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \left(\sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\sin b, \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   8. sin-lowering-sin.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

8. Applied egg-rr 99.4%

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

Alternative 3: 76.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. fmm-def N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \cos \color{blue}{a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\sin b, \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(0 - \sin a\right)\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \sin a\right)\right)\right)\right) \]

   11. sin-lowering-sin.f64 99.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   2. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\sin b, \sin a\right)\right)\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right)\right)\right) \]

   6. sin-lowering-sin.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.4%

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

   1. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)}{\color{blue}{\cos b \cdot \cos a + \sin b \cdot \sin a}}\right)\right) \]

   3. sqr-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right) \cdot \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)}{\cos b \cdot \color{blue}{\cos a} + \sin b \cdot \sin a}\right)\right) \]

   4. difference-of-squares N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\left(\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right) \cdot \left(\cos b \cdot \cos a - \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)}{\color{blue}{\cos b \cdot \cos a} + \sin b \cdot \sin a}\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right) \cdot \left(\cos b \cdot \cos a - \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)}{\color{blue}{\cos b} \cdot \cos a + \sin b \cdot \sin a}\right)\right) \]

   6. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\cos \left(b + a\right) \cdot \left(\cos b \cdot \cos a - \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)}{\color{blue}{\cos b} \cdot \cos a + \sin b \cdot \sin a}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\cos \left(b + a\right) \cdot \left(\cos b \cdot \cos a - \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}{\cos b \cdot \cos a + \sin b \cdot \sin a}\right)\right) \]

   8. cancel-sign-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\cos \left(b + a\right) \cdot \left(\cos b \cdot \cos a + \sin b \cdot \sin a\right)}{\cos b \cdot \color{blue}{\cos a} + \sin b \cdot \sin a}\right)\right) \]

   9. cos-diff N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\cos \left(b + a\right) \cdot \cos \left(b - a\right)}{\cos b \cdot \color{blue}{\cos a} + \sin b \cdot \sin a}\right)\right) \]

   10. cos-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\frac{\cos \left(\left(b + a\right) + \left(b - a\right)\right) + \cos \left(\left(b + a\right) - \left(b - a\right)\right)}{2}}{\color{blue}{\cos b \cdot \cos a} + \sin b \cdot \sin a}\right)\right) \]

   11. cos-diff N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\frac{\frac{\cos \left(\left(b + a\right) + \left(b - a\right)\right) + \cos \left(\left(b + a\right) - \left(b - a\right)\right)}{2}}{\cos \left(b - a\right)}\right)\right) \]

10. Applied egg-rr 76.0%

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

11. Final simplification 76.0%

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

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{if}\;b \leq -240000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ (cos b) (sin b)))))
   (if (<= b -240000.0)
     t_0
     (if (<= b 2.15e-6) (/ (* r (sin b)) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (cos(b) / sin(b));
	double tmp;
	if (b <= -240000.0) {
		tmp = t_0;
	} else if (b <= 2.15e-6) {
		tmp = (r * sin(b)) / 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 / (cos(b) / sin(b))
    if (b <= (-240000.0d0)) then
        tmp = t_0
    else if (b <= 2.15d-6) then
        tmp = (r * sin(b)) / 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.cos(b) / Math.sin(b));
	double tmp;
	if (b <= -240000.0) {
		tmp = t_0;
	} else if (b <= 2.15e-6) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / (math.cos(b) / math.sin(b))
	tmp = 0
	if b <= -240000.0:
		tmp = t_0
	elif b <= 2.15e-6:
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(cos(b) / sin(b)))
	tmp = 0.0
	if (b <= -240000.0)
		tmp = t_0;
	elseif (b <= 2.15e-6)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r / (cos(b) / sin(b));
	tmp = 0.0;
	if (b <= -240000.0)
		tmp = t_0;
	elseif (b <= 2.15e-6)
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -240000.0], t$95$0, If[LessEqual[b, 2.15e-6], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4e5 or 2.15000000000000017e-6 < b

   1. Initial program 55.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos b}{\sin b}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos b, \color{blue}{\sin b}\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(b\right), \sin \color{blue}{b}\right)\right) \]

      3. sin-lowering-sin.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   9. Simplified 56.8%

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

   if -2.4e5 < b < 2.15000000000000017e-6

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 98.0%

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

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{if}\;b \leq -240000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ (cos b) (sin b)))))
   (if (<= b -240000.0) t_0 (if (<= b 2.4e-6) (* (sin b) (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (cos(b) / sin(b));
	double tmp;
	if (b <= -240000.0) {
		tmp = t_0;
	} else if (b <= 2.4e-6) {
		tmp = sin(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 / (cos(b) / sin(b))
    if (b <= (-240000.0d0)) then
        tmp = t_0
    else if (b <= 2.4d-6) then
        tmp = sin(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.cos(b) / Math.sin(b));
	double tmp;
	if (b <= -240000.0) {
		tmp = t_0;
	} else if (b <= 2.4e-6) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / (math.cos(b) / math.sin(b))
	tmp = 0
	if b <= -240000.0:
		tmp = t_0
	elif b <= 2.4e-6:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(cos(b) / sin(b)))
	tmp = 0.0
	if (b <= -240000.0)
		tmp = t_0;
	elseif (b <= 2.4e-6)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r / (cos(b) / sin(b));
	tmp = 0.0;
	if (b <= -240000.0)
		tmp = t_0;
	elseif (b <= 2.4e-6)
		tmp = sin(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[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -240000.0], t$95$0, If[LessEqual[b, 2.4e-6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4e5 or 2.3999999999999999e-6 < b

   1. Initial program 55.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos b}{\sin b}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos b, \color{blue}{\sin b}\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(b\right), \sin \color{blue}{b}\right)\right) \]

      3. sin-lowering-sin.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   9. Simplified 56.8%

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

   if -2.4e5 < b < 2.3999999999999999e-6

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 97.9%

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

       1. associate-/r/ N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos a}\right), \color{blue}{\sin b}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos a\right), \sin \color{blue}{b}\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \sin b\right) \]

       5. sin-lowering-sin.f64 98.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   11. Applied egg-rr 98.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -240000:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((b + a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   8. sin-lowering-sin.f64 75.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 75.7%

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

7. Final simplification 75.7%

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

Alternative 7: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

   8. sin-lowering-sin.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

6. Applied egg-rr 75.6%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
8. Step-by-step derivation

   1. cos-lowering-cos.f64 52.9%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

9. Simplified 52.9%

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

    1. associate-/r/ N/A

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

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos a}\right), \color{blue}{\sin b}\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos a\right), \sin \color{blue}{b}\right) \]

    4. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \sin b\right) \]

    5. sin-lowering-sin.f64 53.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

11. Applied egg-rr 53.0%

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

12. Final simplification 53.0%

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

Alternative 8: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -1.12 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;\frac{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -1.12e+14)
     t_0
     (if (<= b 0.48)
       (/ (* (+ 1.0 (* b (* b -0.16666666666666666))) (* r b)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -1.12e+14) {
		tmp = t_0;
	} else if (b <= 0.48) {
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / cos((b + 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(b)
    if (b <= (-1.12d+14)) then
        tmp = t_0
    else if (b <= 0.48d0) then
        tmp = ((1.0d0 + (b * (b * (-0.16666666666666666d0)))) * (r * b)) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -1.12e+14) {
		tmp = t_0;
	} else if (b <= 0.48) {
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -1.12e+14:
		tmp = t_0
	elif b <= 0.48:
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -1.12e+14)
		tmp = t_0;
	elseif (b <= 0.48)
		tmp = Float64(Float64(Float64(1.0 + Float64(b * Float64(b * -0.16666666666666666))) * Float64(r * b)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -1.12e+14)
		tmp = t_0;
	elseif (b <= 0.48)
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.12e+14], t$95$0, If[LessEqual[b, 0.48], N[(N[(N[(1.0 + N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.12e14 or 0.47999999999999998 < b

   1. Initial program 54.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 54.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 11.3%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 11.3%

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

   if -1.12e14 < b < 0.47999999999999998

   1. Initial program 97.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.4%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(b \cdot \left(r + \frac{-1}{6} \cdot \left({b}^{2} \cdot r\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left(\frac{-1}{6} \cdot \left({b}^{2} \cdot r\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left(\left({b}^{2} \cdot r\right) \cdot \frac{-1}{6} + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left({b}^{2} \cdot \left(r \cdot \frac{-1}{6}\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right)\right) + b \cdot r\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right)\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({b}^{2} \cdot \frac{-1}{6}\right) \cdot r\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot r\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(r \cdot b\right) + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(b \cdot r\right) + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      11. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2} + 1\right) \cdot \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {b}^{2}\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({b}^{2} \cdot \frac{-1}{6}\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(b \cdot b\right) \cdot \frac{-1}{6}\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(b \cdot \left(b \cdot \frac{-1}{6}\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \frac{-1}{6}\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \left(r \cdot b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      21. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(r, b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 96.5%

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

Alternative 9: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -2700000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -2700000000000.0)
     t_0
     (if (<= b 0.48)
       (/ (* r (* b (+ 1.0 (* b (* b -0.16666666666666666))))) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -2700000000000.0) {
		tmp = t_0;
	} else if (b <= 0.48) {
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / cos((b + 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(b)
    if (b <= (-2700000000000.0d0)) then
        tmp = t_0
    else if (b <= 0.48d0) then
        tmp = (r * (b * (1.0d0 + (b * (b * (-0.16666666666666666d0)))))) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -2700000000000.0) {
		tmp = t_0;
	} else if (b <= 0.48) {
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -2700000000000.0:
		tmp = t_0
	elif b <= 0.48:
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -2700000000000.0)
		tmp = t_0;
	elseif (b <= 0.48)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(b * Float64(b * -0.16666666666666666))))) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -2700000000000.0)
		tmp = t_0;
	elseif (b <= 0.48)
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2700000000000.0], t$95$0, If[LessEqual[b, 0.48], N[(N[(r * N[(b * N[(1.0 + N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.7e12 or 0.47999999999999998 < b

   1. Initial program 54.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 54.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 11.3%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 11.3%

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

   if -2.7e12 < b < 0.47999999999999998

   1. Initial program 97.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.4%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {b}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({b}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(b \cdot b\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \left(b \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      7. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 96.5%

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

Alternative 10: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -310000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3200:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -310000.0) t_0 (if (<= b 3200.0) (/ (* r b) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -310000.0) {
		tmp = t_0;
	} else if (b <= 3200.0) {
		tmp = (r * b) / cos((b + 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(b)
    if (b <= (-310000.0d0)) then
        tmp = t_0
    else if (b <= 3200.0d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -310000.0) {
		tmp = t_0;
	} else if (b <= 3200.0) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -310000.0:
		tmp = t_0
	elif b <= 3200.0:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -310000.0)
		tmp = t_0;
	elseif (b <= 3200.0)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -310000.0)
		tmp = t_0;
	elseif (b <= 3200.0)
		tmp = (r * b) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -310000.0], t$95$0, If[LessEqual[b, 3200.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.1e5 or 3200 < b

   1. Initial program 55.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.5%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 11.5%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 11.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 11.4%

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

   if -3.1e5 < b < 3200

   1. Initial program 96.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 96.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 96.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(b \cdot r\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      2. *-lowering-*.f64 95.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

   7. Simplified 95.7%

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

Alternative 11: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -250000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 13.5:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -250000.0) t_0 (if (<= b 13.5) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -250000.0) {
		tmp = t_0;
	} else if (b <= 13.5) {
		tmp = r * (b / 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(b)
    if (b <= (-250000.0d0)) then
        tmp = t_0
    else if (b <= 13.5d0) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -250000.0) {
		tmp = t_0;
	} else if (b <= 13.5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -250000.0:
		tmp = t_0
	elif b <= 13.5:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -250000.0)
		tmp = t_0;
	elseif (b <= 13.5)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -250000.0)
		tmp = t_0;
	elseif (b <= 13.5)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -250000.0], t$95$0, If[LessEqual[b, 13.5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.5e5 or 13.5 < b

   1. Initial program 55.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 55.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 55.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 55.6%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.5%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 11.5%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 11.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 11.4%

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

   if -2.5e5 < b < 13.5

   1. Initial program 96.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 96.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 96.7%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 96.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 96.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{\cos a}\right), \color{blue}{r}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \cos a\right), r\right) \]

      5. cos-lowering-cos.f64 96.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{cos.f64}\left(a\right)\right), r\right) \]

   9. Applied egg-rr 96.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -250000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 13.5:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -310000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 13.5:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -310000.0) t_0 (if (<= b 13.5) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -310000.0) {
		tmp = t_0;
	} else if (b <= 13.5) {
		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(b)
    if (b <= (-310000.0d0)) then
        tmp = t_0
    else if (b <= 13.5d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -310000.0) {
		tmp = t_0;
	} else if (b <= 13.5) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -310000.0:
		tmp = t_0
	elif b <= 13.5:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -310000.0)
		tmp = t_0;
	elseif (b <= 13.5)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -310000.0)
		tmp = t_0;
	elseif (b <= 13.5)
		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[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -310000.0], t$95$0, If[LessEqual[b, 13.5], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.1e5 or 13.5 < b

   1. Initial program 55.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 55.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 55.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

      8. sin-lowering-sin.f64 55.6%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.5%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

   9. Simplified 11.5%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 11.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 11.4%

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

   if -3.1e5 < b < 13.5

   1. Initial program 96.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 96.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 96.7%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 96.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 96.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{r}{\cos a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \color{blue}{\cos a}\right)\right) \]

      5. cos-lowering-cos.f64 96.4%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right)\right) \]

   9. Applied egg-rr 96.4%

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

Alternative 13: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{\cos \left(b + a\right)}{\sin b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos \left(b + a\right), \color{blue}{\sin b}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(b + a\right)\right), \sin \color{blue}{b}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \sin b\right)\right) \]

   8. sin-lowering-sin.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

6. Applied egg-rr 75.6%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{\cos a}, \mathsf{sin.f64}\left(b\right)\right)\right) \]
8. Step-by-step derivation

   1. cos-lowering-cos.f64 52.9%

      \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right)\right) \]

9. Simplified 52.9%

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

10. Taylor expanded in a around 0

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

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

    2. sin-lowering-sin.f64 38.7%

       \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

12. Simplified 38.7%

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

Alternative 14: 34.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 75.6%

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

5. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

   4. cos-lowering-cos.f64 48.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

7. Simplified 48.8%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 34.6%

      \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{b}\right) \]

10. Simplified 34.6%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))