Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 12

Speedup: 1.0×

Specification

\[0 \leq e \land e \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e * N[Cos[v], $MachinePrecision] + 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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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. Final simplification 99.8%

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

Alternative 3: 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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-udef 99.8%

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

   4. associate-/l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in e around 0 99.7%

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

   1. +-commutative 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 98.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(e + 1.0), $MachinePrecision] / N[Sin[v], $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. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.0%

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

5. Final simplification 98.0%

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

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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-udef 99.8%

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

   4. associate-/l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in e around 0 99.7%

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

   1. +-commutative 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in v around 0 98.1%

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

11. Final simplification 98.1%

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

Alternative 6: 97.7% 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 99.8%

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

   1. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in e around 0 97.7%

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

5. Final simplification 97.7%

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

Alternative 7: 52.4% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(0.16666666666666666 * N[(v * N[(e + 1.0), $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. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.0%

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

5. Taylor expanded in v around 0 54.4%

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

6. Final simplification 54.4%

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

Alternative 8: 51.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 / v), $MachinePrecision] + N[(v * 0.16666666666666666), $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. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 54.4%

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

5. Taylor expanded in e around 0 54.0%

   \[\leadsto \color{blue}{\frac{e}{0.16666666666666666 \cdot v + \frac{1}{v}}} \]

6. Final simplification 54.0%

   \[\leadsto \frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \]

Alternative 9: 50.9% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e * 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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around 0 53.4%

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

   1. associate-/l* 53.3%

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

   2. associate-/r/ 53.4%

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

   3. +-commutative 53.4%

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

6. Simplified 53.4%

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

   1. clear-num 53.4%

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

   2. associate-/r/ 53.4%

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

8. Applied egg-rr 53.4%

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

9. Taylor expanded in e around 0 53.2%

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

    1. neg-mul-1 53.2%

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

    2. sub-neg 53.2%

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

11. Simplified 53.2%

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

12. Final simplification 53.2%

    \[\leadsto v \cdot \left(e \cdot \left(1 - e\right)\right) \]

Alternative 10: 51.4% 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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around 0 53.4%

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

   1. associate-/l* 53.3%

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

   2. associate-/r/ 53.4%

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

   3. +-commutative 53.4%

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

6. Simplified 53.4%

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

7. Final simplification 53.4%

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

Alternative 11: 50.3% 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 99.8%

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

   1. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around 0 53.4%

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

   1. associate-/l* 53.3%

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

   2. associate-/r/ 53.4%

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

   3. +-commutative 53.4%

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

6. Simplified 53.4%

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

7. Taylor expanded in e around 0 53.0%

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

8. Final simplification 53.0%

   \[\leadsto e \cdot v \]

Alternative 12: 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. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around 0 53.4%

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

   1. associate-/l* 53.3%

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

   2. associate-/r/ 53.4%

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

   3. +-commutative 53.4%

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

6. Simplified 53.4%

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

   1. clear-num 53.4%

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

   2. associate-/r/ 53.4%

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

8. Applied egg-rr 53.4%

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

9. Taylor expanded in e around inf 4.6%

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

10. Final simplification 4.6%

    \[\leadsto v \]

Reproduce

herbie shell --seed 2023320 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))