Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 9

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

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

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]

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 3: 98.7% 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 99.8%

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

3. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \left(\sin v + -1 \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right)} \]
4. 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. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.4

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

5. Applied rewrites 98.4%

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

Alternative 4: 98.8% 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 99.8%

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

3. Taylor expanded in v around 0

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

   1. lower-+.f64 98.3

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

5. Applied rewrites 98.3%

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

Alternative 5: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[(1.0 - e), $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 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. *-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. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.8%

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

Alternative 6: 97.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. *-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 97.6

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

5. Applied rewrites 97.6%

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

Alternative 7: 50.7% 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 50.9

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

5. Applied rewrites 50.9%

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

Alternative 8: 50.1% 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 99.8%

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

3. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \left(\sin v + -1 \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right)} \]
4. 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. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 50.5%

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

Alternative 9: 49.6% 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 50.9

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))