Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

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, 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 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 99.9%

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

   1. *-commutative 99.9%

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

   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 3: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return ((e * sin(v)) / (1.0 - (e * e))) * (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 * e))) * (1.0d0 - e)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in v around 0 99.3%

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

   1. flip-+ 99.3%

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

   2. associate-/r/ 99.3%

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

   3. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.4%

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

   3. mul-1-neg 99.4%

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

   4. fma-neg 99.4%

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

   5. associate-*r* 99.4%

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

   6. *-commutative 99.4%

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

   7. unpow2 99.4%

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

6. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out-- 99.4%

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

   3. *-commutative 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in v around 0 99.2%

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

    1. unpow2 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 5: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in v around 0 99.3%

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

3. Final simplification 99.3%

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

Alternative 6: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   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 98.7%

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

5. Final simplification 98.7%

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

Alternative 7: 51.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in v around 0 99.3%

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

   1. clear-num 98.6%

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

   2. inv-pow 98.6%

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

   3. associate-/r* 98.3%

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

   4. +-commutative 98.3%

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

4. Applied egg-rr 98.3%

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

   1. unpow-1 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in v around 0 54.4%

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

8. Final simplification 54.4%

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

Alternative 8: 51.0% 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 99.9%

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

   1. *-commutative 99.9%

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

   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 53.8%

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

   1. associate-/l* 53.7%

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

   2. +-commutative 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in e around 0 53.7%

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

   1. associate-*r* 53.7%

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

   2. neg-mul-1 53.7%

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

   3. distribute-rgt-out 53.7%

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

   4. +-commutative 53.7%

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

   5. sub-neg 53.7%

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

   6. unpow2 53.7%

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

9. Simplified 53.7%

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

10. Final simplification 53.7%

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

Alternative 9: 51.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   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 53.8%

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

5. Final simplification 53.8%

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

Alternative 10: 50.5% accurate, 69.7× 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 99.9%

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

   1. *-commutative 99.9%

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

   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 53.8%

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

   1. associate-/l* 53.7%

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

   2. +-commutative 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in e around 0 53.3%

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

   1. *-commutative 53.3%

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

9. Simplified 53.3%

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

10. Final simplification 53.3%

    \[\leadsto e \cdot v \]

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 99.9%

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

   1. *-commutative 99.9%

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

   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 53.8%

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

   1. associate-/l* 53.7%

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

   2. +-commutative 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in e around inf 4.5%

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

8. Final simplification 4.5%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023279 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))