rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 11.7s

Alternatives: 15

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: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-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 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. lift-*.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 4: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\cos \left(a - b\right)}{0.5} \cdot \frac{\sin b \cdot r}{\cos \left(b - \left(a - \left(a + b\right)\right)\right) + \cos \left(\left(\left(b - a\right) - a\right) - b\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  (/ (cos (- a b)) 0.5)
  (/ (* (sin b) r) (+ (cos (- b (- a (+ a b)))) (cos (- (- (- b a) a) b))))))

C

double code(double r, double a, double b) {
	return (cos((a - b)) / 0.5) * ((sin(b) * r) / (cos((b - (a - (a + b)))) + cos((((b - a) - 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 = (cos((a - b)) / 0.5d0) * ((sin(b) * r) / (cos((b - (a - (a + b)))) + cos((((b - a) - a) - b))))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.cos((a - b)) / 0.5) * ((Math.sin(b) * r) / (Math.cos((b - (a - (a + b)))) + Math.cos((((b - a) - a) - b))));
}

Python

def code(r, a, b):
	return (math.cos((a - b)) / 0.5) * ((math.sin(b) * r) / (math.cos((b - (a - (a + b)))) + math.cos((((b - a) - a) - b))))

Julia

function code(r, a, b)
	return Float64(Float64(cos(Float64(a - b)) / 0.5) * Float64(Float64(sin(b) * r) / Float64(cos(Float64(b - Float64(a - Float64(a + b)))) + cos(Float64(Float64(Float64(b - a) - a) - b)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (cos((a - b)) / 0.5) * ((sin(b) * r) / (cos((b - (a - (a + b)))) + cos((((b - a) - a) - b))));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Cos[N[(a - b), $MachinePrecision]], $MachinePrecision] / 0.5), $MachinePrecision] * N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[Cos[N[(b - N[(a - N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(N[(b - a), $MachinePrecision] - a), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 99.6

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 76.2%

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

7. Final simplification 76.2%

   \[\leadsto \frac{\cos \left(a - b\right)}{0.5} \cdot \frac{\sin b \cdot r}{\cos \left(b - \left(a - \left(a + b\right)\right)\right) + \cos \left(\left(\left(b - a\right) - a\right) - b\right)} \]
8. Add Preprocessing

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -6.2e-5)
     (/ t_0 (cos b))
     (if (<= b 5.8e-10) (/ t_0 (cos a)) (* (/ r (cos b)) (sin b))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -6.2e-5) {
		tmp = t_0 / cos(b);
	} else if (b <= 5.8e-10) {
		tmp = t_0 / cos(a);
	} else {
		tmp = (r / cos(b)) * sin(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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (b <= (-6.2d-5)) then
        tmp = t_0 / cos(b)
    else if (b <= 5.8d-10) then
        tmp = t_0 / cos(a)
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (b <= -6.2e-5) {
		tmp = t_0 / Math.cos(b);
	} else if (b <= 5.8e-10) {
		tmp = t_0 / Math.cos(a);
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if b <= -6.2e-5:
		tmp = t_0 / math.cos(b)
	elif b <= 5.8e-10:
		tmp = t_0 / math.cos(a)
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -6.2e-5)
		tmp = Float64(t_0 / cos(b));
	elseif (b <= 5.8e-10)
		tmp = Float64(t_0 / cos(a));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (b <= -6.2e-5)
		tmp = t_0 / cos(b);
	elseif (b <= 5.8e-10)
		tmp = t_0 / cos(a);
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -6.2e-5], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-10], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000027e-5

   1. Initial program 48.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 48.8

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

   5. Applied rewrites 48.8%

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

   if -6.20000000000000027e-5 < b < 5.79999999999999962e-10

   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

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

      1. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.79999999999999962e-10 < b

   1. Initial program 58.0%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.4

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

   5. Applied rewrites 58.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00039:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 310000000:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.00039)
   (/ (* (sin b) r) (cos a))
   (if (<= a 310000000.0)
     (* (/ r (cos b)) (sin b))
     (* (/ r (cos a)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00039) {
		tmp = (sin(b) * r) / cos(a);
	} else if (a <= 310000000.0) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (r / cos(a)) * sin(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 (a <= (-0.00039d0)) then
        tmp = (sin(b) * r) / cos(a)
    else if (a <= 310000000.0d0) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00039) {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	} else if (a <= 310000000.0) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.00039:
		tmp = (math.sin(b) * r) / math.cos(a)
	elif a <= 310000000.0:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.00039)
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	elseif (a <= 310000000.0)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.00039)
		tmp = (sin(b) * r) / cos(a);
	elseif (a <= 310000000.0)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.00039], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 310000000.0], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.89999999999999993e-4

   1. Initial program 56.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -3.89999999999999993e-4 < a < 3.1e8

   1. Initial program 97.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 3.1e8 < a

   1. Initial program 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-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 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 45.2

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

   7. Applied rewrites 45.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow-1 N/A

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

      12. remove-double-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 45.2

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

   9. Applied rewrites 45.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00039:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 310000000:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -0.000108:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -0.000108) t_0 (if (<= b 5.8e-10) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -0.000108) {
		tmp = t_0;
	} else if (b <= 5.8e-10) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r / cos(b)) * sin(b)
    if (b <= (-0.000108d0)) then
        tmp = t_0
    else if (b <= 5.8d-10) then
        tmp = (b * r) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (b <= -0.000108) {
		tmp = t_0;
	} else if (b <= 5.8e-10) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if b <= -0.000108:
		tmp = t_0
	elif b <= 5.8e-10:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -0.000108)
		tmp = t_0;
	elseif (b <= 5.8e-10)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (b <= -0.000108)
		tmp = t_0;
	elseif (b <= 5.8e-10)
		tmp = (b * r) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.000108], t$95$0, If[LessEqual[b, 5.8e-10], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.08e-4 or 5.79999999999999962e-10 < b

   1. Initial program 53.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 54.0

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

   5. Applied rewrites 54.0%

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

   if -1.08e-4 < b < 5.79999999999999962e-10

   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 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

Alternative 8: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Final simplification 75.9%

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

Alternative 9: 76.8% 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 75.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-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 75.9

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

4. Applied rewrites 75.9%

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

Alternative 10: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 72:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -7.2e+17)
     t_0
     (if (<= b 72.0)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -7.2e+17) {
		tmp = t_0;
	} else if (b <= 72.0) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -7.2e+17)
		tmp = t_0;
	elseif (b <= 72.0)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -7.2e+17], t$95$0, If[LessEqual[b, 72.0], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.2e17 or 72 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-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 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.3

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 12.3%

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

   if -7.2e17 < b < 72

   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 97.8%

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

Alternative 11: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3100:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -1.6e+18) t_0 (if (<= b 3100.0) (/ (* b r) (cos (+ a b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -1.6e+18) {
		tmp = t_0;
	} else if (b <= 3100.0) {
		tmp = (b * r) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / 1.0d0) * r
    if (b <= (-1.6d+18)) then
        tmp = t_0
    else if (b <= 3100.0d0) then
        tmp = (b * r) / cos((a + b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / 1.0) * r;
	double tmp;
	if (b <= -1.6e+18) {
		tmp = t_0;
	} else if (b <= 3100.0) {
		tmp = (b * r) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / 1.0) * r
	tmp = 0
	if b <= -1.6e+18:
		tmp = t_0
	elif b <= 3100.0:
		tmp = (b * r) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -1.6e+18)
		tmp = t_0;
	elseif (b <= 3100.0)
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / 1.0) * r;
	tmp = 0.0;
	if (b <= -1.6e+18)
		tmp = t_0;
	elseif (b <= 3100.0)
		tmp = (b * r) / cos((a + b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -1.6e+18], t$95$0, If[LessEqual[b, 3100.0], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6e18 or 3100 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-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 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.3

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 12.3%

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

   if -1.6e18 < b < 3100

   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 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 12: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3900:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -68.0) t_0 (if (<= b 3900.0) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -68.0) {
		tmp = t_0;
	} else if (b <= 3900.0) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / 1.0d0) * r
    if (b <= (-68.0d0)) then
        tmp = t_0
    else if (b <= 3900.0d0) then
        tmp = (b * r) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / 1.0) * r;
	double tmp;
	if (b <= -68.0) {
		tmp = t_0;
	} else if (b <= 3900.0) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / 1.0) * r
	tmp = 0
	if b <= -68.0:
		tmp = t_0
	elif b <= 3900.0:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -68.0)
		tmp = t_0;
	elseif (b <= 3900.0)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / 1.0) * r;
	tmp = 0.0;
	if (b <= -68.0)
		tmp = t_0;
	elseif (b <= 3900.0)
		tmp = (b * r) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -68.0], t$95$0, If[LessEqual[b, 3900.0], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -68 or 3900 < b

   1. Initial program 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-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 52.8

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.2

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

   7. Applied rewrites 11.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 12.1%

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

   if -68 < b < 3900

   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 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.0%

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

Alternative 13: 51.2% 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 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 50.3

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

5. Applied rewrites 50.3%

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

7. Applied rewrites 50.3%

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

Alternative 14: 51.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 50.3

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

5. Applied rewrites 50.3%

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

Alternative 15: 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 75.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 50.3

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 37.4%

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

Reproduce

herbie shell --seed 2024241 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))