rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 11.5s

Alternatives: 10

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

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

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 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

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

   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 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

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

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + (-N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $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-/.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 75.9

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

   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 75.9

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

4. Applied rewrites 75.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. unsub-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0095)
   (/ (* r (sin b)) (cos b))
   (if (<= b 0.0195)
     (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (cos (+ b a)))
     (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0095) {
		tmp = (r * sin(b)) / cos(b);
	} else if (b <= 0.0195) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / cos((b + a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0095)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (b <= 0.0195)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / cos(Float64(b + a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0095], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0195], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00949999999999999976

   1. Initial program 41.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 42.9

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

   5. Applied rewrites 42.9%

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

   if -0.00949999999999999976 < b < 0.0195

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + \frac{-1}{6} \cdot \left({b}^{2} \cdot r\right)\right)}}{\cos \left(a + b\right)} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.0195 < b

   1. Initial program 57.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.0

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

      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 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.8

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

   7. Applied rewrites 58.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -0.0095)
     t_0
     (if (<= b 0.0195)
       (/ (* 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) / cos(b));
	double tmp;
	if (b <= -0.0095) {
		tmp = t_0;
	} else if (b <= 0.0195) {
		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(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -0.0095)
		tmp = t_0;
	elseif (b <= 0.0195)
		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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0095], t$95$0, If[LessEqual[b, 0.0195], 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 < -0.00949999999999999976 or 0.0195 < b

   1. Initial program 48.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 48.5

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

      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 48.5

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 49.9

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

   7. Applied rewrites 49.9%

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

   if -0.00949999999999999976 < b < 0.0195

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + \frac{-1}{6} \cdot \left({b}^{2} \cdot r\right)\right)}}{\cos \left(a + b\right)} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Final simplification 75.9%

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

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

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

   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 75.9

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

4. Applied rewrites 75.9%

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

5. Final simplification 75.9%

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

Alternative 7: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 75.9

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

   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 75.9

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

4. Applied rewrites 75.9%

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

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 58.9

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

7. Applied rewrites 58.9%

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

8. Final simplification 58.9%

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

Alternative 8: 51.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Final simplification 55.6%

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

Alternative 9: 51.1% 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 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. 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 55.5

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

5. Applied rewrites 55.5%

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

7. Applied rewrites 55.6%

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

Alternative 10: 35.1% 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 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. 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 55.5

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

5. Applied rewrites 55.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))