Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 12

Speedup: 1.0×

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos v, e, -1\right) \cdot \frac{e \cdot \sin v}{{\left(\cos v \cdot e\right)}^{2} - 1} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (* (fma (cos v) e -1.0) (/ (* e (sin v)) (- (pow (* (cos v) e) 2.0) 1.0))))

C

double code(double e, double v) {
	return fma(cos(v), e, -1.0) * ((e * sin(v)) / (pow((cos(v) * e), 2.0) - 1.0));
}

Julia

function code(e, v)
	return Float64(fma(cos(v), e, -1.0) * Float64(Float64(e * sin(v)) / Float64((Float64(cos(v) * e) ^ 2.0) - 1.0)))
end

Wolfram

code[e_, v_] := N[(N[(N[Cos[v], $MachinePrecision] * e + -1.0), $MachinePrecision] * N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision], 2.0], $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

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(\cos v, e, -1\right) \cdot \frac{e \cdot \sin v}{{\left(\cos v \cdot e\right)}^{2} - 1} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in e around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + -1 \cdot \left(e \cdot \cos v\right)\right) \cdot e\right)} \cdot \sin v \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + -1 \cdot \left(e \cdot \cos v\right)\right) \cdot e\right)} \cdot \sin v \]

   3. mul-1-neg N/A

      \[\leadsto \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(e \cdot \cos v\right)\right)}\right) \cdot e\right) \cdot \sin v \]

   4. unsub-neg N/A

      \[\leadsto \left(\color{blue}{\left(1 - e \cdot \cos v\right)} \cdot e\right) \cdot \sin v \]

   5. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(1 - e \cdot \cos v\right)} \cdot e\right) \cdot \sin v \]

   6. *-commutative N/A

      \[\leadsto \left(\left(1 - \color{blue}{\cos v \cdot e}\right) \cdot e\right) \cdot \sin v \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\left(1 - \color{blue}{\cos v \cdot e}\right) \cdot e\right) \cdot \sin v \]

   8. lower-cos.f64 99.3

      \[\leadsto \left(\left(1 - \color{blue}{\cos v} \cdot e\right) \cdot e\right) \cdot \sin v \]

7. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left(\left(1 - \cos v \cdot e\right) \cdot e\right)} \cdot \sin v \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[(1.0 - N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision]), $MachinePrecision] * N[Sin[v], $MachinePrecision]), $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 v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 52.1%

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

9. Taylor expanded in e around 0

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(\sin v + -1 \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right) \cdot e} \]

    2. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\sin v + -1 \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right) \cdot e} \]

    3. associate-*r* N/A

       \[\leadsto \left(\sin v + -1 \cdot \color{blue}{\left(\left(e \cdot \cos v\right) \cdot \sin v\right)}\right) \cdot e \]

    4. associate-*r* N/A

       \[\leadsto \left(\sin v + \color{blue}{\left(-1 \cdot \left(e \cdot \cos v\right)\right) \cdot \sin v}\right) \cdot e \]

    5. distribute-rgt1-in N/A

       \[\leadsto \color{blue}{\left(\left(-1 \cdot \left(e \cdot \cos v\right) + 1\right) \cdot \sin v\right)} \cdot e \]

    6. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot \left(e \cdot \cos v\right)\right)} \cdot \sin v\right) \cdot e \]

    7. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(1 + -1 \cdot \left(e \cdot \cos v\right)\right) \cdot \sin v\right)} \cdot e \]

    8. mul-1-neg N/A

       \[\leadsto \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(e \cdot \cos v\right)\right)}\right) \cdot \sin v\right) \cdot e \]

    9. unsub-neg N/A

       \[\leadsto \left(\color{blue}{\left(1 - e \cdot \cos v\right)} \cdot \sin v\right) \cdot e \]

    10. lower--.f64 N/A

        \[\leadsto \left(\color{blue}{\left(1 - e \cdot \cos v\right)} \cdot \sin v\right) \cdot e \]

    11. *-commutative N/A

        \[\leadsto \left(\left(1 - \color{blue}{\cos v \cdot e}\right) \cdot \sin v\right) \cdot e \]

    12. lower-*.f64 N/A

        \[\leadsto \left(\left(1 - \color{blue}{\cos v \cdot e}\right) \cdot \sin v\right) \cdot e \]

    13. lower-cos.f64 N/A

        \[\leadsto \left(\left(1 - \color{blue}{\cos v} \cdot e\right) \cdot \sin v\right) \cdot e \]

    14. lower-sin.f64 99.3

        \[\leadsto \left(\left(1 - \cos v \cdot e\right) \cdot \color{blue}{\sin v}\right) \cdot e \]

11. Applied rewrites 99.3%

    \[\leadsto \color{blue}{\left(\left(1 - \cos v \cdot e\right) \cdot \sin v\right) \cdot e} \]
12. Add Preprocessing

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

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

   1. lower-+.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 6: 97.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[Sin[v], $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 e around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 97.8

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

5. Applied rewrites 97.8%

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

6. Final simplification 97.8%

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

Alternative 7: 51.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right)\\ \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot e - \mathsf{fma}\left(t\_0, -0.16666666666666666, \mathsf{fma}\left(0.008333333333333333, e, 0.008333333333333333\right)\right), v \cdot v, t\_0\right), v \cdot v, 1 + e\right)}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (fma -0.5 e (fma 0.16666666666666666 e 0.16666666666666666))))
   (/
    e
    (/
     (fma
      (fma
       (-
        (* 0.041666666666666664 e)
        (fma
         t_0
         -0.16666666666666666
         (fma 0.008333333333333333 e 0.008333333333333333)))
       (* v v)
       t_0)
      (* v v)
      (+ 1.0 e))
     v))))

C

double code(double e, double v) {
	double t_0 = fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666));
	return e / (fma(fma(((0.041666666666666664 * e) - fma(t_0, -0.16666666666666666, fma(0.008333333333333333, e, 0.008333333333333333))), (v * v), t_0), (v * v), (1.0 + e)) / v);
}

Julia

function code(e, v)
	t_0 = fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666))
	return Float64(e / Float64(fma(fma(Float64(Float64(0.041666666666666664 * e) - fma(t_0, -0.16666666666666666, fma(0.008333333333333333, e, 0.008333333333333333))), Float64(v * v), t_0), Float64(v * v), Float64(1.0 + e)) / v))
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(-0.5 * e + N[(0.16666666666666666 * e + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(N[(N[(0.041666666666666664 * e), $MachinePrecision] - N[(t$95$0 * -0.16666666666666666 + N[(0.008333333333333333 * e + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(1.0 + e), $MachinePrecision]), $MachinePrecision] / v), $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. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{e}{\color{blue}{\frac{1 + \left(e + {v}^{2} \cdot \left(\left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{1}{24} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + \left(e + {v}^{2} \cdot \left(\left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{1}{24} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}}} \]

7. Applied rewrites 53.9%

   \[\leadsto \frac{e}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot e - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right), -0.16666666666666666, \mathsf{fma}\left(0.008333333333333333, e, 0.008333333333333333\right)\right), v \cdot v, \mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right)\right), v \cdot v, 1 + e\right)}{v}}} \]
8. Add Preprocessing

Alternative 8: 52.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right), v \cdot v, 1 + e\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (/
   (fma
    (fma -0.5 e (fma 0.16666666666666666 e 0.16666666666666666))
    (* v v)
    (+ 1.0 e))
   v)))

C

double code(double e, double v) {
	return e / (fma(fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666)), (v * v), (1.0 + e)) / v);
}

Julia

function code(e, v)
	return Float64(e / Float64(fma(fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666)), Float64(v * v), Float64(1.0 + e)) / v))
end

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(-0.5 * e + N[(0.16666666666666666 * e + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(1.0 + e), $MachinePrecision]), $MachinePrecision] / v), $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. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-+.f64 53.8

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

7. Applied rewrites 53.8%

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

Alternative 9: 50.8% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e / N[(1.0 + e), $MachinePrecision]), $MachinePrecision] * v), $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. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 52.7

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

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{e}{1 + e} \cdot v} \]
6. Add Preprocessing

Alternative 10: 50.4% accurate, 11.8× speedup

Math

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

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(Float64(-e) * v) * e))
end

Wolfram

code[e_, v_] := N[(v * e + N[(N[((-e) * v), $MachinePrecision] * e), $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. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 52.1%

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

9. Taylor expanded in e around 0

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

11. Applied rewrites 52.5%

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

13. Applied rewrites 52.5%

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

Alternative 11: 50.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v - N[(e * v), $MachinePrecision]), $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 v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 52.7

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

5. Applied rewrites 52.7%

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

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 52.5%

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

Alternative 12: 49.9% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * v), $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. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 52.1%

   \[\leadsto e \cdot \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))