Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

Alternatives: 9

Speedup: N/A×

Specification

\[0 \leq e \land e \leq 1\]

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (/ (sin v) (fma e (cos v) 1.0))))

C

double code(double e, double v) {
	return e * (sin(v) / fma(e, cos(v), 1.0));
}

Julia

function code(e, v)
	return Float64(e * Float64(sin(v) / fma(e, cos(v), 1.0)))
end

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot \sin v}}{1 + e \cdot \cos v} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   6. lower-/.f64 99.8

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v}} \cdot e \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{1 + e \cdot \cos v}} \cdot e \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v} + 1} \cdot e \]

   10. lower-fma.f64 99.8

       \[\leadsto \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]

5. Final simplification 99.8%

   \[\leadsto e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \]
6. Add Preprocessing

Alternative 2: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{\sin v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (/ (sin v) (+ e 1.0))))

C

double code(double e, double v) {
	return e * (sin(v) / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (sin(v) / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return e * (Math.sin(v) / (e + 1.0));
}

Python

def code(e, v):
	return e * (math.sin(v) / (e + 1.0))

Julia

function code(e, v)
	return Float64(e * Float64(sin(v) / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (sin(v) / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot \sin v}}{1 + e \cdot \cos v} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   6. lower-/.f64 99.8

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v}} \cdot e \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{1 + e \cdot \cos v}} \cdot e \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v} + 1} \cdot e \]

   10. lower-fma.f64 99.8

       \[\leadsto \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\sin v}{\color{blue}{1 + e}} \cdot e \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e + 1}} \cdot e \]

   2. lower-+.f64 99.1

      \[\leadsto \frac{\sin v}{\color{blue}{e + 1}} \cdot e \]

7. Applied rewrites 99.1%

   \[\leadsto \frac{\sin v}{\color{blue}{e + 1}} \cdot e \]

8. Final simplification 99.1%

   \[\leadsto e \cdot \frac{\sin v}{e + 1} \]
9. Add Preprocessing

Alternative 3: 97.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sin v \cdot e \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (sin v) e))

C

double code(double e, double v) {
	return sin(v) * e;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) * e
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) * e;
}

Python

def code(e, v):
	return math.sin(v) * e

Julia

function code(e, v)
	return Float64(sin(v) * e)
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) * e;
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \sin v} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{e \cdot \sin v} \]

   2. lower-sin.f64 98.4

      \[\leadsto e \cdot \color{blue}{\sin v} \]

5. Applied rewrites 98.4%

   \[\leadsto \color{blue}{e \cdot \sin v} \]

6. Final simplification 98.4%

   \[\leadsto \sin v \cdot e \]
7. Add Preprocessing

Alternative 4: 51.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{v \cdot e}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot 0.041666666666666664, -0.5\right)\right), e + 1\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  (* v e)
  (fma v (* v (* e (fma v (* v 0.041666666666666664) -0.5))) (+ e 1.0))))

C

double code(double e, double v) {
	return (v * e) / fma(v, (v * (e * fma(v, (v * 0.041666666666666664), -0.5))), (e + 1.0));
}

Julia

function code(e, v)
	return Float64(Float64(v * e) / fma(v, Float64(v * Float64(e * fma(v, Float64(v * 0.041666666666666664), -0.5))), Float64(e + 1.0)))
end

Wolfram

code[e_, v_] := N[(N[(v * e), $MachinePrecision] / N[(v * N[(v * N[(e * N[(v * N[(v * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e}} \]
4. Step-by-step derivation

   1. lower-+.f64 99.1

      \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e}} \]

5. Applied rewrites 99.1%

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e}} \]

6. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{v \cdot \left(e + \frac{-1}{6} \cdot \left(e \cdot {v}^{2}\right)\right)}}{1 + e} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{v \cdot \left(e + \frac{-1}{6} \cdot \left(e \cdot {v}^{2}\right)\right)}}{1 + e} \]

   2. +-commutative N/A

      \[\leadsto \frac{v \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(e \cdot {v}^{2}\right) + e\right)}}{1 + e} \]

   3. *-commutative N/A

      \[\leadsto \frac{v \cdot \left(\color{blue}{\left(e \cdot {v}^{2}\right) \cdot \frac{-1}{6}} + e\right)}{1 + e} \]

   4. associate-*l* N/A

      \[\leadsto \frac{v \cdot \left(\color{blue}{e \cdot \left({v}^{2} \cdot \frac{-1}{6}\right)} + e\right)}{1 + e} \]

   5. *-commutative N/A

      \[\leadsto \frac{v \cdot \left(e \cdot \color{blue}{\left(\frac{-1}{6} \cdot {v}^{2}\right)} + e\right)}{1 + e} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{v \cdot \color{blue}{\mathsf{fma}\left(e, \frac{-1}{6} \cdot {v}^{2}, e\right)}}{1 + e} \]

   7. *-commutative N/A

      \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \color{blue}{{v}^{2} \cdot \frac{-1}{6}}, e\right)}{1 + e} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \color{blue}{{v}^{2} \cdot \frac{-1}{6}}, e\right)}{1 + e} \]

   9. unpow2 N/A

      \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \color{blue}{\left(v \cdot v\right)} \cdot \frac{-1}{6}, e\right)}{1 + e} \]

   10. lower-*.f64 53.7

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \color{blue}{\left(v \cdot v\right)} \cdot -0.16666666666666666, e\right)}{1 + e} \]

8. Applied rewrites 53.7%

   \[\leadsto \frac{\color{blue}{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot -0.16666666666666666, e\right)}}{1 + e} \]

9. Taylor expanded in v around 0

   \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. associate-+r+ N/A

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{\left(1 + e\right) + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right)}} \]

    2. +-commutative N/A

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{{v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right) + \left(1 + e\right)}} \]

    3. unpow2 N/A

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{\left(v \cdot v\right)} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right) + \left(1 + e\right)} \]

    4. associate-*l* N/A

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{v \cdot \left(v \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right)\right)} + \left(1 + e\right)} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot \frac{-1}{6}, e\right)}{\color{blue}{\mathsf{fma}\left(v, v \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right), 1 + e\right)}} \]

11. Applied rewrites 53.7%

    \[\leadsto \frac{v \cdot \mathsf{fma}\left(e, \left(v \cdot v\right) \cdot -0.16666666666666666, e\right)}{\color{blue}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot 0.041666666666666664, -0.5\right)\right), e + 1\right)}} \]

12. Taylor expanded in v around 0

    \[\leadsto \frac{\color{blue}{e \cdot v}}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot \frac{1}{24}, \frac{-1}{2}\right)\right), e + 1\right)} \]
13. Step-by-step derivation

    1. lower-*.f64 54.8

       \[\leadsto \frac{\color{blue}{e \cdot v}}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot 0.041666666666666664, -0.5\right)\right), e + 1\right)} \]

14. Applied rewrites 54.8%

    \[\leadsto \frac{\color{blue}{e \cdot v}}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot 0.041666666666666664, -0.5\right)\right), e + 1\right)} \]

15. Final simplification 54.8%

    \[\leadsto \frac{v \cdot e}{\mathsf{fma}\left(v, v \cdot \left(e \cdot \mathsf{fma}\left(v, v \cdot 0.041666666666666664, -0.5\right)\right), e + 1\right)} \]
16. Add Preprocessing

Alternative 5: 50.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \frac{v \cdot e}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* v e) (+ e 1.0)))

C

double code(double e, double v) {
	return (v * e) / (e + 1.0);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (v * e) / (e + 1.0d0)
end function

Java

public static double code(double e, double v) {
	return (v * e) / (e + 1.0);
}

Python

def code(e, v):
	return (v * e) / (e + 1.0)

Julia

function code(e, v)
	return Float64(Float64(v * e) / Float64(e + 1.0))
end

MATLAB

function tmp = code(e, v)
	tmp = (v * e) / (e + 1.0);
end

Wolfram

code[e_, v_] := N[(N[(v * e), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + e} \]

   3. lower-+.f64 53.9

      \[\leadsto \frac{e \cdot v}{\color{blue}{1 + e}} \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

6. Final simplification 53.9%

   \[\leadsto \frac{v \cdot e}{e + 1} \]
7. Add Preprocessing

Alternative 6: 50.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, e, -e \cdot \left(v \cdot e\right)\right) \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (fma v e (- (* e (* v e)))))

C

double code(double e, double v) {
	return fma(v, e, -(e * (v * e)));
}

Julia

function code(e, v)
	return fma(v, e, Float64(-Float64(e * Float64(v * e))))
end

Wolfram

code[e_, v_] := N[(v * e + (-N[(e * N[(v * e), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + e} \]

   3. lower-+.f64 53.9

      \[\leadsto \frac{e \cdot v}{\color{blue}{1 + e}} \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

6. Taylor expanded in e around 0

   \[\leadsto e \cdot \color{blue}{\left(v + -1 \cdot \left(e \cdot v\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 53.7%

   \[\leadsto e \cdot \color{blue}{\left(v - e \cdot v\right)} \]
9. Step-by-step derivation

10. Applied rewrites 53.8%

    \[\leadsto \mathsf{fma}\left(v, e, \left(v \cdot \left(-e\right)\right) \cdot e\right) \]

11. Final simplification 53.8%

    \[\leadsto \mathsf{fma}\left(v, e, -e \cdot \left(v \cdot e\right)\right) \]
12. Add Preprocessing

Alternative 7: 50.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ e \cdot \left(v \cdot \left(1 - e\right)\right) \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (* v (- 1.0 e))))

C

double code(double e, double v) {
	return e * (v * (1.0 - e));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v * (1.0d0 - e))
end function

Java

public static double code(double e, double v) {
	return e * (v * (1.0 - e));
}

Python

def code(e, v):
	return e * (v * (1.0 - e))

Julia

function code(e, v)
	return Float64(e * Float64(v * Float64(1.0 - e)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v * (1.0 - e));
end

Wolfram

code[e_, v_] := N[(e * N[(v * N[(1.0 - e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + e} \]

   3. lower-+.f64 53.9

      \[\leadsto \frac{e \cdot v}{\color{blue}{1 + e}} \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

6. Taylor expanded in e around 0

   \[\leadsto e \cdot \color{blue}{\left(v + -1 \cdot \left(e \cdot v\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 53.7%

   \[\leadsto e \cdot \color{blue}{\left(v - e \cdot v\right)} \]
9. Step-by-step derivation

10. Applied rewrites 53.7%

    \[\leadsto e \cdot \left(\left(\left(-e\right) + 1\right) \cdot v\right) \]

11. Final simplification 53.7%

    \[\leadsto e \cdot \left(v \cdot \left(1 - e\right)\right) \]
12. Add Preprocessing

Alternative 8: 50.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ e \cdot \left(v - v \cdot e\right) \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (- v (* v e))))

C

double code(double e, double v) {
	return e * (v - (v * e));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v - (v * e))
end function

Java

public static double code(double e, double v) {
	return e * (v - (v * e));
}

Python

def code(e, v):
	return e * (v - (v * e))

Julia

function code(e, v)
	return Float64(e * Float64(v - Float64(v * e)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v - (v * e));
end

Wolfram

code[e_, v_] := N[(e * N[(v - N[(v * e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + e} \]

   3. lower-+.f64 53.9

      \[\leadsto \frac{e \cdot v}{\color{blue}{1 + e}} \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

6. Taylor expanded in e around 0

   \[\leadsto e \cdot \color{blue}{\left(v + -1 \cdot \left(e \cdot v\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 53.7%

   \[\leadsto e \cdot \color{blue}{\left(v - e \cdot v\right)} \]

9. Final simplification 53.7%

   \[\leadsto e \cdot \left(v - v \cdot e\right) \]
10. Add Preprocessing

Alternative 9: 49.7% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ v \cdot e \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* v e))

C

double code(double e, double v) {
	return v * e;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v * e
end function

Java

public static double code(double e, double v) {
	return v * e;
}

Python

def code(e, v):
	return v * e

Julia

function code(e, v)
	return Float64(v * e)
end

MATLAB

function tmp = code(e, v)
	tmp = v * e;
end

Wolfram

code[e_, v_] := N[(v * e), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + e} \]

   3. lower-+.f64 53.9

      \[\leadsto \frac{e \cdot v}{\color{blue}{1 + e}} \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.3%

   \[\leadsto e \cdot \color{blue}{v} \]

9. Final simplification 53.3%

   \[\leadsto v \cdot e \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))