rsin A (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 11.1s

Alternatives: 13

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 4: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -23.5 \lor \neg \left(b \leq 4.4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -23.5) (not (<= b 4.4e-6)))
   (* (/ r (cos b)) (sin b))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -23.5) || !(b <= 4.4e-6)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (b * r) / 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 ((b <= (-23.5d0)) .or. (.not. (b <= 4.4d-6))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -23.5) || !(b <= 4.4e-6)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -23.5) or not (b <= 4.4e-6):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -23.5) || !(b <= 4.4e-6))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -23.5) || ~((b <= 4.4e-6)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -23.5], N[Not[LessEqual[b, 4.4e-6]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -23.5 or 4.4000000000000002e-6 < b

   1. Initial program 52.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 52.4

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

   5. Applied rewrites 52.4%

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

   if -23.5 < b < 4.4000000000000002e-6

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -23.5 \lor \neg \left(b \leq 4.4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -23.5:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -23.5)
   (* (/ r (cos b)) (sin b))
   (if (<= b 4.4e-6) (/ (* b r) (cos a)) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -23.5) {
		tmp = (r / cos(b)) * sin(b);
	} else if (b <= 4.4e-6) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-23.5d0)) then
        tmp = (r / cos(b)) * sin(b)
    else if (b <= 4.4d-6) then
        tmp = (b * r) / cos(a)
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -23.5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else if (b <= 4.4e-6) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -23.5:
		tmp = (r / math.cos(b)) * math.sin(b)
	elif b <= 4.4e-6:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -23.5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	elseif (b <= 4.4e-6)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -23.5)
		tmp = (r / cos(b)) * sin(b);
	elseif (b <= 4.4e-6)
		tmp = (b * r) / cos(a);
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -23.5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-6], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -23.5

   1. Initial program 44.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 44.2

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

   5. Applied rewrites 44.2%

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

   if -23.5 < b < 4.4000000000000002e-6

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   if 4.4000000000000002e-6 < b

   1. Initial program 58.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.0

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

   5. Applied rewrites 58.0%

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

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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]

Derivation

1. Initial program 74.0%

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

Alternative 7: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 74.0

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

4. Applied rewrites 74.0%

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

Alternative 8: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15.2 \lor \neg \left(b \leq 3.1\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -15.2) (not (<= b 3.1)))
   (/ (* r (sin b)) 1.0)
   (/
    (*
     (*
      r
      (fma
       (fma 0.008333333333333333 (* b b) -0.16666666666666666)
       (* b b)
       1.0))
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -15.2) || !(b <= 3.1)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -15.2) || !(b <= 3.1))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -15.2], N[Not[LessEqual[b, 3.1]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -15.199999999999999 or 3.10000000000000009 < b

   1. Initial program 51.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.7%

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

   if -15.199999999999999 < b < 3.10000000000000009

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -15.2 \lor \neg \left(b \leq 3.1\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15.2 \lor \neg \left(b \leq 3.1\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -15.2) (not (<= b 3.1)))
   (/ (* r (sin b)) 1.0)
   (/ (* b r) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -15.2) || !(b <= 3.1)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (b * r) / cos((a + b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-15.2d0)) .or. (.not. (b <= 3.1d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = (b * r) / cos((a + b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -15.2) || !(b <= 3.1)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = (b * r) / Math.cos((a + b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -15.2) or not (b <= 3.1):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = (b * r) / math.cos((a + b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -15.2) || !(b <= 3.1))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -15.2) || ~((b <= 3.1)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = (b * r) / cos((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -15.2], N[Not[LessEqual[b, 3.1]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -15.199999999999999 or 3.10000000000000009 < b

   1. Initial program 51.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.7%

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

   if -15.199999999999999 < b < 3.10000000000000009

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -15.2 \lor \neg \left(b \leq 3.1\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11 \lor \neg \left(b \leq 1.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -11.0) (not (<= b 1.6)))
   (/ (* r (sin b)) 1.0)
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -11.0) || !(b <= 1.6)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (b * r) / 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 ((b <= (-11.0d0)) .or. (.not. (b <= 1.6d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -11.0) || !(b <= 1.6)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -11.0) or not (b <= 1.6):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -11.0) || !(b <= 1.6))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -11.0) || ~((b <= 1.6)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -11.0], N[Not[LessEqual[b, 1.6]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -11 or 1.6000000000000001 < b

   1. Initial program 51.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.7%

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

   if -11 < b < 1.6000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11 \lor \neg \left(b \leq 1.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 48.0

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

5. Applied rewrites 48.0%

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

7. Applied rewrites 48.0%

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

Alternative 12: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 48.0

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

5. Applied rewrites 48.0%

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

7. Applied rewrites 48.0%

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

Alternative 13: 34.3% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 48.0

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

5. Applied rewrites 48.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024309 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))