rsin A (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 13.3s

Alternatives: 17

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: 76.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. cos-sum N/A

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 77.4%

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

4. Applied egg-rr 77.4%

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

   1. cos-sum N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= a -4.2e-6)
     (/ t_0 (cos a))
     (if (<= a 1650.0) (/ t_0 (cos b)) (* r (/ (sin b) (cos a)))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (a <= -4.2e-6) {
		tmp = t_0 / cos(a);
	} else if (a <= 1650.0) {
		tmp = t_0 / cos(b);
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (a <= (-4.2d-6)) then
        tmp = t_0 / cos(a)
    else if (a <= 1650.0d0) then
        tmp = t_0 / cos(b)
    else
        tmp = r * (sin(b) / cos(a))
    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 (a <= -4.2e-6) {
		tmp = t_0 / Math.cos(a);
	} else if (a <= 1650.0) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if a <= -4.2e-6:
		tmp = t_0 / math.cos(a)
	elif a <= 1650.0:
		tmp = t_0 / math.cos(b)
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (a <= -4.2e-6)
		tmp = Float64(t_0 / cos(a));
	elseif (a <= 1650.0)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (a <= -4.2e-6)
		tmp = t_0 / cos(a);
	elseif (a <= 1650.0)
		tmp = t_0 / cos(b);
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.2e-6], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1650.0], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1999999999999996e-6

   1. Initial program 57.2%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 59.1%

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

   5. Simplified 59.1%

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

   if -4.1999999999999996e-6 < a < 1650

   1. Initial program 98.1%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. cos-lowering-cos.f64 97.8%

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

   5. Simplified 97.8%

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

   if 1650 < a

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 54.0%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 54.3%

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

   7. Simplified 54.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -3.65e-6)
   (/ (* r (sin b)) (cos a))
   (if (<= a 1650.0) (* r (/ (sin b) (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.65e-6) {
		tmp = (r * sin(b)) / cos(a);
	} else if (a <= 1650.0) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.65d-6)) then
        tmp = (r * sin(b)) / cos(a)
    else if (a <= 1650.0d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.65e-6) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else if (a <= 1650.0) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -3.65e-6:
		tmp = (r * math.sin(b)) / math.cos(a)
	elif a <= 1650.0:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -3.65e-6)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	elseif (a <= 1650.0)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -3.65e-6)
		tmp = (r * sin(b)) / cos(a);
	elseif (a <= 1650.0)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -3.65e-6], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1650.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.65000000000000021e-6

   1. Initial program 57.2%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 59.1%

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

   5. Simplified 59.1%

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

   if -3.65000000000000021e-6 < a < 1650

   1. Initial program 98.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 97.7%

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

   7. Simplified 97.7%

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

   if 1650 < a

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 54.0%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 54.3%

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

   7. Simplified 54.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -4.2e-6) t_0 (if (<= a 1650.0) (* r (/ (sin b) (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -4.2e-6) {
		tmp = t_0;
	} else if (a <= 1650.0) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / cos(a))
    if (a <= (-4.2d-6)) then
        tmp = t_0
    else if (a <= 1650.0d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / Math.cos(a));
	double tmp;
	if (a <= -4.2e-6) {
		tmp = t_0;
	} else if (a <= 1650.0) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -4.2e-6:
		tmp = t_0
	elif a <= 1650.0:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -4.2e-6)
		tmp = t_0;
	elseif (a <= 1650.0)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -4.2e-6)
		tmp = t_0;
	elseif (a <= 1650.0)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.2e-6], t$95$0, If[LessEqual[a, 1650.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1999999999999996e-6 or 1650 < a

   1. Initial program 55.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 55.6%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 56.7%

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

   7. Simplified 56.7%

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

   if -4.1999999999999996e-6 < a < 1650

   1. Initial program 98.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 97.7%

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

   7. Simplified 97.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Final simplification 77.5%

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

Alternative 8: 76.8% 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 77.5%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. sin-lowering-sin.f64 77.5%

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

4. Applied egg-rr 77.5%

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

5. Final simplification 77.5%

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

Alternative 9: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 77.4%

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

4. Applied egg-rr 77.4%

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

5. Taylor expanded in b around 0

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

   1. cos-lowering-cos.f64 54.4%

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

7. Simplified 54.4%

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

8. Final simplification 54.4%

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

Alternative 10: 55.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -3.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 61:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -3.3)
     t_0
     (if (<= b 61.0)
       (/
        (*
         r
         (*
          b
          (+
           1.0
           (*
            (* b b)
            (+ -0.16666666666666666 (* (* b b) 0.008333333333333333))))))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -3.3) {
		tmp = t_0;
	} else if (b <= 61.0) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / 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 <= (-3.3d0)) then
        tmp = t_0
    else if (b <= 61.0d0) then
        tmp = (r * (b * (1.0d0 + ((b * b) * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0)))))) / 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 <= -3.3) {
		tmp = t_0;
	} else if (b <= 61.0) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / 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 <= -3.3:
		tmp = t_0
	elif b <= 61.0:
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / 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 <= -3.3)
		tmp = t_0;
	elseif (b <= 61.0)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(Float64(b * b) * Float64(-0.16666666666666666 + Float64(Float64(b * b) * 0.008333333333333333)))))) / 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 <= -3.3)
		tmp = t_0;
	elseif (b <= 61.0)
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / 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, -3.3], t$95$0, If[LessEqual[b, 61.0], N[(N[(r * N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $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 < -3.2999999999999998 or 61 < b

   1. Initial program 57.0%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.4%

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

   8. Simplified 12.4%

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

   if -3.2999999999999998 < b < 61

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({b}^{2}\right), \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{120} \cdot {b}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{-1}{6} + \frac{1}{120} \cdot {b}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {b}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({b}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({b}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(b \cdot b\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

      13. *-lowering-*.f64 98.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right) \]

   5. Simplified 98.0%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 61:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -270:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1560000:\\ \;\;\;\;\frac{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -270.0)
     t_0
     (if (<= b 1560000.0)
       (/ (* (+ 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 <= -270.0) {
		tmp = t_0;
	} else if (b <= 1560000.0) {
		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 <= (-270.0d0)) then
        tmp = t_0
    else if (b <= 1560000.0d0) 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 <= -270.0) {
		tmp = t_0;
	} else if (b <= 1560000.0) {
		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 <= -270.0:
		tmp = t_0
	elif b <= 1560000.0:
		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 <= -270.0)
		tmp = t_0;
	elseif (b <= 1560000.0)
		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 <= -270.0)
		tmp = t_0;
	elseif (b <= 1560000.0)
		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, -270.0], t$95$0, If[LessEqual[b, 1560000.0], 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 < -270 or 1.56e6 < b

   1. Initial program 56.3%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.5%

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

   8. Simplified 12.5%

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

   if -270 < b < 1.56e6

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -270:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1560000:\\ \;\;\;\;\frac{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -380:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1800000:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -380.0)
     t_0
     (if (<= b 1800000.0)
       (/ (* 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 <= -380.0) {
		tmp = t_0;
	} else if (b <= 1800000.0) {
		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 <= (-380.0d0)) then
        tmp = t_0
    else if (b <= 1800000.0d0) 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 <= -380.0) {
		tmp = t_0;
	} else if (b <= 1800000.0) {
		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 <= -380.0:
		tmp = t_0
	elif b <= 1800000.0:
		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 <= -380.0)
		tmp = t_0;
	elseif (b <= 1800000.0)
		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 <= -380.0)
		tmp = t_0;
	elseif (b <= 1800000.0)
		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, -380.0], t$95$0, If[LessEqual[b, 1800000.0], 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 < -380 or 1.8e6 < b

   1. Initial program 56.3%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.5%

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

   8. Simplified 12.5%

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

   if -380 < b < 1.8e6

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -380:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1800000:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -380:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1800000:\\ \;\;\;\;\left(r \cdot b\right) \cdot \frac{1 + \left(b \cdot b\right) \cdot -0.16666666666666666}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -380.0)
     t_0
     (if (<= b 1800000.0)
       (* (* 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 <= -380.0) {
		tmp = t_0;
	} else if (b <= 1800000.0) {
		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 <= (-380.0d0)) then
        tmp = t_0
    else if (b <= 1800000.0d0) 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 <= -380.0) {
		tmp = t_0;
	} else if (b <= 1800000.0) {
		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 <= -380.0:
		tmp = t_0
	elif b <= 1800000.0:
		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 <= -380.0)
		tmp = t_0;
	elseif (b <= 1800000.0)
		tmp = Float64(Float64(r * b) * Float64(Float64(1.0 + Float64(Float64(b * 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 <= -380.0)
		tmp = t_0;
	elseif (b <= 1800000.0)
		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, -380.0], t$95$0, If[LessEqual[b, 1800000.0], N[(N[(r * b), $MachinePrecision] * N[(N[(1.0 + N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -380 or 1.8e6 < b

   1. Initial program 56.3%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.5%

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

   8. Simplified 12.5%

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

   if -380 < b < 1.8e6

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 95.7%

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

   5. Simplified 95.7%

      \[\leadsto \frac{\color{blue}{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. +-lowering-+.f64 95.6%

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -380:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1800000:\\ \;\;\;\;\left(r \cdot b\right) \cdot \frac{1 + \left(b \cdot b\right) \cdot -0.16666666666666666}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -2.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1400:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -2.85) t_0 (if (<= b 1400.0) (/ (* 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 <= -2.85) {
		tmp = t_0;
	} else if (b <= 1400.0) {
		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 <= (-2.85d0)) then
        tmp = t_0
    else if (b <= 1400.0d0) 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 <= -2.85) {
		tmp = t_0;
	} else if (b <= 1400.0) {
		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 <= -2.85:
		tmp = t_0
	elif b <= 1400.0:
		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 <= -2.85)
		tmp = t_0;
	elseif (b <= 1400.0)
		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 <= -2.85)
		tmp = t_0;
	elseif (b <= 1400.0)
		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, -2.85], t$95$0, If[LessEqual[b, 1400.0], 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 < -2.85000000000000009 or 1400 < b

   1. Initial program 57.0%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.4%

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

   8. Simplified 12.4%

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

   if -2.85000000000000009 < b < 1400

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.85:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1400:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1560000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -1.3) t_0 (if (<= b 1560000.0) (* 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 <= -1.3) {
		tmp = t_0;
	} else if (b <= 1560000.0) {
		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 <= (-1.3d0)) then
        tmp = t_0
    else if (b <= 1560000.0d0) 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 <= -1.3) {
		tmp = t_0;
	} else if (b <= 1560000.0) {
		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 <= -1.3:
		tmp = t_0
	elif b <= 1560000.0:
		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 <= -1.3)
		tmp = t_0;
	elseif (b <= 1560000.0)
		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 <= -1.3)
		tmp = t_0;
	elseif (b <= 1560000.0)
		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, -1.3], t$95$0, If[LessEqual[b, 1560000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.30000000000000004 or 1.56e6 < b

   1. Initial program 56.9%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. sin-lowering-sin.f64 12.5%

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

   8. Simplified 12.5%

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

   if -1.30000000000000004 < b < 1.56e6

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. /-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 96.5%

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

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{r \cdot b}{\cos a}} \]
   6. 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 96.6%

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1560000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.6% 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 77.5%

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

3. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

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

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

   6. sin-lowering-sin.f64 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

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

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

   3. sin-lowering-sin.f64 37.2%

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

8. Simplified 37.2%

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

9. Final simplification 37.2%

   \[\leadsto r \cdot \sin b \]
10. Add Preprocessing

Alternative 17: 34.6% 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 77.5%

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

3. Taylor expanded in b around 0

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

   1. /-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 49.6%

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

5. Simplified 49.6%

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

6. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 32.9%

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

8. Simplified 32.9%

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

9. Final simplification 32.9%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))