Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 12.0s

Alternatives: 11

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e \cdot \cos v\\ \frac{\sin v \cdot e}{1 - {t_0}^{2}} \cdot \left(1 - t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (* e (cos v))))
   (* (/ (* (sin v) e) (- 1.0 (pow t_0 2.0))) (- 1.0 t_0))))

C

double code(double e, double v) {
	double t_0 = e * cos(v);
	return ((sin(v) * e) / (1.0 - pow(t_0, 2.0))) * (1.0 - t_0);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    t_0 = e * cos(v)
    code = ((sin(v) * e) / (1.0d0 - (t_0 ** 2.0d0))) * (1.0d0 - t_0)
end function

Java

public static double code(double e, double v) {
	double t_0 = e * Math.cos(v);
	return ((Math.sin(v) * e) / (1.0 - Math.pow(t_0, 2.0))) * (1.0 - t_0);
}

Python

def code(e, v):
	t_0 = e * math.cos(v)
	return ((math.sin(v) * e) / (1.0 - math.pow(t_0, 2.0))) * (1.0 - t_0)

Julia

function code(e, v)
	t_0 = Float64(e * cos(v))
	return Float64(Float64(Float64(sin(v) * e) / Float64(1.0 - (t_0 ^ 2.0))) * Float64(1.0 - t_0))
end

MATLAB

function tmp = code(e, v)
	t_0 = e * cos(v);
	tmp = ((sin(v) * e) / (1.0 - (t_0 ^ 2.0))) * (1.0 - t_0);
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-/l* 99.8%

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

   4. *-commutative 99.8%

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

   5. flip-+ 99.8%

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

   6. associate-/r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. metadata-eval 99.8%

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

   9. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-/l* 99.8%

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

   4. *-commutative 99.8%

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

   5. flip-+ 99.8%

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

   6. associate-/r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. metadata-eval 99.8%

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

   9. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v \cdot e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\right)\right)} \]

   2. expm1-udef 30.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin v \cdot e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\right)} - 1} \]

   3. associate-*l/ 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(\sin v \cdot e\right) \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}}}\right)} - 1 \]

   4. associate-/l* 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin v \cdot e}{\frac{1 - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}}\right)} - 1 \]

   5. *-commutative 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{e \cdot \sin v}}{\frac{1 - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}\right)} - 1 \]

   6. metadata-eval 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\frac{\color{blue}{1 \cdot 1} - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}\right)} - 1 \]

   7. unpow2 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\frac{1 \cdot 1 - \color{blue}{\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}}{1 - e \cdot \cos v}}\right)} - 1 \]

   8. flip-+ 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\color{blue}{1 + e \cdot \cos v}}\right)} - 1 \]

   9. +-commutative 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\color{blue}{e \cdot \cos v + 1}}\right)} - 1 \]

7. Applied egg-rr 30.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{e \cdot \cos v + 1}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e \cdot \sin v}{e \cdot \cos v + 1}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l* 99.6%

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

   4. fma-def 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in e around 0 99.6%

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

    1. expm1-log1p-u 99.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \frac{e \cdot \cos v}{\sin v}}\right)\right)} \]

    2. expm1-udef 30.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \frac{e \cdot \cos v}{\sin v}}\right)} - 1} \]

    3. associate-/l* 30.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \color{blue}{\frac{e}{\frac{\sin v}{\cos v}}}}\right)} - 1 \]

    4. quot-tan 30.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \frac{e}{\color{blue}{\tan v}}}\right)} - 1 \]

12. Applied egg-rr 30.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \frac{e}{\tan v}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 99.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e}{\frac{1}{\sin v} + \frac{e}{\tan v}}\right)\right)} \]

    2. expm1-log1p 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

    \[\leadsto \frac{e}{\frac{1}{\sin v} + \frac{e}{\tan v}} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $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. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in e around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 98.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.6%

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

   1. +-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in v around inf 98.8%

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

   1. +-commutative 98.8%

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

   2. *-commutative 98.8%

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

   3. /-rgt-identity 98.8%

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

   4. associate-/r/ 98.6%

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

   5. associate-/r* 98.6%

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

   6. distribute-rgt-in 98.7%

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

   7. rgt-mult-inverse 98.7%

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

   8. *-lft-identity 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 97.9% accurate, 2.0× 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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in e around 0 97.9%

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

5. Final simplification 97.9%

   \[\leadsto \sin v \cdot e \]

Alternative 7: 52.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e / ((v * ((e * (-0.5d0)) + ((-0.16666666666666666d0) * ((-1.0d0) - e)))) + ((1.0d0 / v) + (e / v)))
end function

Java

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

Python

def code(e, v):
	return e / ((v * ((e * -0.5) + (-0.16666666666666666 * (-1.0 - e)))) + ((1.0 / v) + (e / v)))

Julia

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

MATLAB

function tmp = code(e, v)
	tmp = e / ((v * ((e * -0.5) + (-0.16666666666666666 * (-1.0 - e)))) + ((1.0 / v) + (e / v)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-/l* 99.8%

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

   4. *-commutative 99.8%

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

   5. flip-+ 99.8%

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

   6. associate-/r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. metadata-eval 99.8%

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

   9. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v \cdot e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\right)\right)} \]

   2. expm1-udef 30.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin v \cdot e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\right)} - 1} \]

   3. associate-*l/ 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(\sin v \cdot e\right) \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}}}\right)} - 1 \]

   4. associate-/l* 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin v \cdot e}{\frac{1 - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}}\right)} - 1 \]

   5. *-commutative 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{e \cdot \sin v}}{\frac{1 - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}\right)} - 1 \]

   6. metadata-eval 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\frac{\color{blue}{1 \cdot 1} - {\left(e \cdot \cos v\right)}^{2}}{1 - e \cdot \cos v}}\right)} - 1 \]

   7. unpow2 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\frac{1 \cdot 1 - \color{blue}{\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}}{1 - e \cdot \cos v}}\right)} - 1 \]

   8. flip-+ 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\color{blue}{1 + e \cdot \cos v}}\right)} - 1 \]

   9. +-commutative 30.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{\color{blue}{e \cdot \cos v + 1}}\right)} - 1 \]

7. Applied egg-rr 30.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e \cdot \sin v}{e \cdot \cos v + 1}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e \cdot \sin v}{e \cdot \cos v + 1}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l* 99.6%

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

   4. fma-def 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in v around 0 52.0%

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

11. Final simplification 52.0%

    \[\leadsto \frac{e}{v \cdot \left(e \cdot -0.5 + -0.16666666666666666 \cdot \left(-1 - e\right)\right) + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

Alternative 8: 51.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 51.2%

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

   1. associate-/l* 51.1%

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

   2. associate-/r/ 51.2%

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

   3. +-commutative 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in e around 0 50.6%

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

   1. mul-1-neg 50.6%

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

   2. unsub-neg 50.6%

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

   3. unpow2 50.6%

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

9. Simplified 50.6%

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

10. Final simplification 50.6%

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

Alternative 9: 51.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 51.2%

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

   1. associate-/l* 51.1%

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

   2. associate-/r/ 51.2%

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

   3. +-commutative 51.2%

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

6. Simplified 51.2%

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

7. Final simplification 51.2%

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

Alternative 10: 50.9% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 51.2%

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

   1. associate-/l* 51.1%

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

   2. associate-/r/ 51.2%

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

   3. +-commutative 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in e around 0 50.3%

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

8. Final simplification 50.3%

   \[\leadsto v \cdot e \]

Alternative 11: 4.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 v)

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

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

Wolfram

code[e_, v_] := v

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 51.2%

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

   1. associate-/l* 51.1%

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

   2. associate-/r/ 51.2%

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

   3. +-commutative 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in e around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023280 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))