Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 15

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.7

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

   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. *-commutative N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.7

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

   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. *-commutative N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in e around 0

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

   1. *-lft-identity N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 97.6

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

7. Applied rewrites 97.6%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r* N/A

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

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

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 97.6

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

5. Applied rewrites 97.6%

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

6. Final simplification 97.6%

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

Alternative 5: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 97.5

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

5. Applied rewrites 97.5%

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

Alternative 6: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.7

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

   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. *-commutative N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-+.f64 97.5

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

7. Applied rewrites 97.5%

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

Alternative 7: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r* N/A

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

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

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 97.6

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 96.7%

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

9. Final simplification 96.7%

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

Alternative 8: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.095:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, v \cdot v, -0.16666666666666666\right), v \cdot v, 1\right) \cdot e\right) \cdot v}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, v \cdot v, 0.041666666666666664\right), v \cdot v, -0.5\right), v \cdot v, 1\right), e, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 0.095)
   (/
    (*
     (*
      (fma (fma 0.008333333333333333 (* v v) -0.16666666666666666) (* v v) 1.0)
      e)
     v)
    (fma
     (fma
      (fma
       (fma -0.001388888888888889 (* v v) 0.041666666666666664)
       (* v v)
       -0.5)
      (* v v)
      1.0)
     e
     1.0))
   (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 0.095) {
		tmp = ((fma(fma(0.008333333333333333, (v * v), -0.16666666666666666), (v * v), 1.0) * e) * v) / fma(fma(fma(fma(-0.001388888888888889, (v * v), 0.041666666666666664), (v * v), -0.5), (v * v), 1.0), e, 1.0);
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 0.095)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(v * v), -0.16666666666666666), Float64(v * v), 1.0) * e) * v) / fma(fma(fma(fma(-0.001388888888888889, Float64(v * v), 0.041666666666666664), Float64(v * v), -0.5), Float64(v * v), 1.0), e, 1.0));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 0.095], N[(N[(N[(N[(N[(0.008333333333333333 * N[(v * v), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * e), $MachinePrecision] * v), $MachinePrecision] / N[(N[(N[(N[(-0.001388888888888889 * N[(v * v), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(v * v), $MachinePrecision] + -0.5), $MachinePrecision] * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * e + 1.0), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 0.095000000000000001

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

      \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\color{blue}{1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) - \frac{1}{2}\right)}, e, 1\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\color{blue}{{v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) - \frac{1}{2}\right) + 1}, e, 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\color{blue}{\left({v}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) - \frac{1}{2}\right) \cdot {v}^{2}} + 1, e, 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({v}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) - \frac{1}{2}, {v}^{2}, 1\right)}, e, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{v}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {v}^{2}, 1\right), e, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) \cdot {v}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {v}^{2}, 1\right), e, 1\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}\right) \cdot {v}^{2} + \color{blue}{\frac{-1}{2}}, {v}^{2}, 1\right), e, 1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {v}^{2}, {v}^{2}, \frac{-1}{2}\right)}, {v}^{2}, 1\right), e, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {v}^{2} + \frac{1}{24}}, {v}^{2}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {v}^{2}, \frac{1}{24}\right)}, {v}^{2}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{v \cdot v}, \frac{1}{24}\right), {v}^{2}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{v \cdot v}, \frac{1}{24}\right), {v}^{2}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), \color{blue}{v \cdot v}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), \color{blue}{v \cdot v}, \frac{-1}{2}\right), {v}^{2}, 1\right), e, 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), v \cdot v, \frac{-1}{2}\right), \color{blue}{v \cdot v}, 1\right), e, 1\right)} \]

      15. lower-*.f64 67.8

          \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, v \cdot v, 0.041666666666666664\right), v \cdot v, -0.5\right), \color{blue}{v \cdot v}, 1\right), e, 1\right)} \]

   7. Applied rewrites 67.8%

      \[\leadsto \frac{\sin v \cdot e}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, v \cdot v, 0.041666666666666664\right), v \cdot v, -0.5\right), v \cdot v, 1\right)}, e, 1\right)} \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{\color{blue}{v \cdot \left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), v \cdot v, \frac{-1}{2}\right), v \cdot v, 1\right), e, 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right) \cdot v}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), v \cdot v, \frac{-1}{2}\right), v \cdot v, 1\right), e, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right) \cdot v}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, v \cdot v, \frac{1}{24}\right), v \cdot v, \frac{-1}{2}\right), v \cdot v, 1\right), e, 1\right)} \]

   10. Applied rewrites 65.8%

       \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, v \cdot v, -0.16666666666666666\right), v \cdot v, 1\right) \cdot e\right) \cdot v}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, v \cdot v, 0.041666666666666664\right), v \cdot v, -0.5\right), v \cdot v, 1\right), e, 1\right)} \]

   if 0.095000000000000001 < v

   1. Initial program 99.4%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\sin v \cdot e} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.095:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, v \cdot v, -0.16666666666666666\right), v \cdot v, 1\right) \cdot e\right) \cdot v}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, v \cdot v, 0.041666666666666664\right), v \cdot v, -0.5\right), v \cdot v, 1\right), e, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.3% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.5

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

   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 99.5

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

4. Applied rewrites 99.5%

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

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 52.6

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

7. Applied rewrites 52.6%

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

Alternative 10: 52.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, e, 0.16666666666666666\right), v \cdot v, e\right) + 1}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (/
   (+ (fma (fma -0.3333333333333333 e 0.16666666666666666) (* v v) e) 1.0)
   v)))

C

double code(double e, double v) {
	return e / ((fma(fma(-0.3333333333333333, e, 0.16666666666666666), (v * v), e) + 1.0) / v);
}

Julia

function code(e, v)
	return Float64(e / Float64(Float64(fma(fma(-0.3333333333333333, e, 0.16666666666666666), Float64(v * v), e) + 1.0) / v))
end

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(N[(-0.3333333333333333 * e + 0.16666666666666666), $MachinePrecision] * N[(v * v), $MachinePrecision] + e), $MachinePrecision] + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.5

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

   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 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 52.5%

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

8. Taylor expanded in v around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

10. Applied rewrites 52.6%

    \[\leadsto \frac{e}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, e, 0.16666666666666666\right), v \cdot v, e\right) + 1}{v}}} \]
11. Add Preprocessing

Alternative 11: 52.0% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 64.4

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

5. Applied rewrites 64.4%

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

6. Taylor expanded in v around 0

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

   1. lower-*.f64 52.1

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

8. Applied rewrites 52.1%

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

Alternative 12: 51.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 51.5

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

5. Applied rewrites 51.5%

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

7. Applied rewrites 51.5%

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

Alternative 13: 51.2% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 51.5

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

5. Applied rewrites 51.5%

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

Alternative 14: 50.7% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 51.5

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

5. Applied rewrites 51.5%

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

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 50.7%

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

Alternative 15: 50.2% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 51.5

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

5. Applied rewrites 51.5%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 49.9%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))