rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 11.2s

Alternatives: 15

Speedup: 0.4×

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: 76.3% 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

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

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 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-neg.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / ((cos(b) * cos(a)) - (sin(b) * sin(a)));
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[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -8.4e-7) t_0 (if (<= b 5.1e-5) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -8.4e-7) {
		tmp = t_0;
	} else if (b <= 5.1e-5) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * (r / cos(b))
    if (b <= (-8.4d-7)) then
        tmp = t_0
    else if (b <= 5.1d-5) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * (r / Math.cos(b));
	double tmp;
	if (b <= -8.4e-7) {
		tmp = t_0;
	} else if (b <= 5.1e-5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(b))
	tmp = 0
	if b <= -8.4e-7:
		tmp = t_0
	elif b <= 5.1e-5:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(b)))
	tmp = 0.0
	if (b <= -8.4e-7)
		tmp = t_0;
	elseif (b <= 5.1e-5)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(b));
	tmp = 0.0;
	if (b <= -8.4e-7)
		tmp = t_0;
	elseif (b <= 5.1e-5)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.4e-7], t$95$0, If[LessEqual[b, 5.1e-5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.4e-7 or 5.09999999999999996e-5 < b

   1. Initial program 59.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if -8.4e-7 < b < 5.09999999999999996e-5

   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 1.0× speedup

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 79.0

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 79.0

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

4. Applied rewrites 79.0%

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

5. Final simplification 79.0%

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

Alternative 5: 76.3% 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 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 79.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 79.0

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

4. Applied rewrites 79.0%

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

5. Final simplification 79.0%

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

Alternative 6: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 64.5

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

5. Applied rewrites 64.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 64.5

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in b around 0

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

   1. lower-cos.f64 55.2

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

10. Applied rewrites 55.2%

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

Alternative 7: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 115:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -5.4)
     t_0
     (if (<= b 115.0)
       (/
        (*
         r
         (fma
          (fma
           b
           (* b (fma b (* b -0.0001984126984126984) 0.008333333333333333))
           -0.16666666666666666)
          (* b (* b b))
          b))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -5.4) {
		tmp = t_0;
	} else if (b <= 115.0) {
		tmp = (r * fma(fma(b, (b * fma(b, (b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -5.4)
		tmp = t_0;
	elseif (b <= 115.0)
		tmp = Float64(Float64(r * fma(fma(b, Float64(b * fma(b, Float64(b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -5.4], t$95$0, If[LessEqual[b, 115.0], N[(N[(r * N[(N[(b * N[(b * N[(b * N[(b * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.4000000000000004 or 115 < b

   1. Initial program 58.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.1%

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

   if -5.4000000000000004 < b < 115

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) + b \cdot 1\right)}}{\cos \left(a + b\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{r \cdot \left(\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)} + b \cdot 1\right)}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right)} + b \cdot 1\right)}{\cos \left(a + b\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}\right)}{\cos \left(a + b\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, b \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.6%

      \[\leadsto \frac{r \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 115:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -4.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -4.4)
     t_0
     (if (<= b 40.0)
       (/
        (*
         b
         (fma
          (* b b)
          (* r (fma (* b b) 0.008333333333333333 -0.16666666666666666))
          r))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -4.4) {
		tmp = t_0;
	} else if (b <= 40.0) {
		tmp = (b * fma((b * b), (r * fma((b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -4.4)
		tmp = t_0;
	elseif (b <= 40.0)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -4.4], t$95$0, If[LessEqual[b, 40.0], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.4000000000000004 or 40 < b

   1. Initial program 58.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.1%

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

   if -4.4000000000000004 < b < 40

   1. Initial program 99.0%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 97.4

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -5.4)
     t_0
     (if (<= b 2.6)
       (/ (* r (fma (* b b) (* b -0.16666666666666666) b)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -5.4) {
		tmp = t_0;
	} else if (b <= 2.6) {
		tmp = (r * fma((b * b), (b * -0.16666666666666666), b)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -5.4)
		tmp = t_0;
	elseif (b <= 2.6)
		tmp = Float64(Float64(r * fma(Float64(b * b), Float64(b * -0.16666666666666666), b)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -5.4], t$95$0, If[LessEqual[b, 2.6], N[(N[(r * N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.4000000000000004 or 2.60000000000000009 < b

   1. Initial program 58.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.0%

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

   if -5.4000000000000004 < b < 2.60000000000000009

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.9

         \[\leadsto \frac{r \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -5.4)
     t_0
     (if (<= b 2.6)
       (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -5.4) {
		tmp = t_0;
	} else if (b <= 2.6) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -5.4)
		tmp = t_0;
	elseif (b <= 2.6)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -5.4], t$95$0, If[LessEqual[b, 2.6], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.4000000000000004 or 2.60000000000000009 < b

   1. Initial program 58.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.0%

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

   if -5.4000000000000004 < b < 2.60000000000000009

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 97.9

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \color{blue}{\left(b \cdot b\right)}, r\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -3.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -3.8) t_0 (if (<= b 40.0) (/ (* r b) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -3.8) {
		tmp = t_0;
	} else if (b <= 40.0) {
		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)) / 1.0d0
    if (b <= (-3.8d0)) then
        tmp = t_0
    else if (b <= 40.0d0) 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)) / 1.0;
	double tmp;
	if (b <= -3.8) {
		tmp = t_0;
	} else if (b <= 40.0) {
		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)) / 1.0
	tmp = 0
	if b <= -3.8:
		tmp = t_0
	elif b <= 40.0:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -3.8)
		tmp = t_0;
	elseif (b <= 40.0)
		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)) / 1.0;
	tmp = 0.0;
	if (b <= -3.8)
		tmp = t_0;
	elseif (b <= 40.0)
		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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -3.8], t$95$0, If[LessEqual[b, 40.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.7999999999999998 or 40 < b

   1. Initial program 58.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.1%

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

   if -3.7999999999999998 < b < 40

   1. Initial program 99.0%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{1}\\ \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.182:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) 1.0)))
   (if (<= b -4.8) t_0 (if (<= b 0.182) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / 1.0;
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 0.182) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * sin(b)) / 1.0d0
    if (b <= (-4.8d0)) then
        tmp = t_0
    else if (b <= 0.182d0) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r * Math.sin(b)) / 1.0;
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 0.182) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / 1.0
	tmp = 0
	if b <= -4.8:
		tmp = t_0
	elif b <= 0.182:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / 1.0)
	tmp = 0.0
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 0.182)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / 1.0;
	tmp = 0.0;
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 0.182)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -4.8], t$95$0, If[LessEqual[b, 0.182], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 0.182 < b

   1. Initial program 58.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.1%

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

   if -4.79999999999999982 < b < 0.182

   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{elif}\;b \leq 0.182:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 51.0

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

5. Applied rewrites 51.0%

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

7. Applied rewrites 51.0%

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

8. Final simplification 51.0%

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

Alternative 14: 50.3% accurate, 1.9× speedup

Math

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

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(b * Float64(r / cos(a)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 51.0

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

5. Applied rewrites 51.0%

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

7. Applied rewrites 51.0%

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

Alternative 15: 33.8% accurate, 36.7× 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 79.0%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))