Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

Alternatives: 10

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. Final simplification 99.8%

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

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 99.8%

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

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

   3. +-commutative 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in e around 0 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * 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-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 98.8%

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

   1. +-commutative 98.8%

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

6. Simplified 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 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 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in e around 0 98.0%

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

5. Final simplification 98.0%

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

Alternative 5: 52.9% accurate, 12.3× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return e / ((v * ((e * -0.5) - -0.16666666666666666)) + ((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 / v) + (e / v)))
end function

Java

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

Python

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

Julia

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

Wolfram

code[e_, v_] := N[(e / N[(N[(v * N[(N[(e * -0.5), $MachinePrecision] - -0.16666666666666666), $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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

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. associate-/l* 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around 0 60.1%

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

8. Taylor expanded in e around 0 60.1%

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

9. Final simplification 60.1%

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

Alternative 6: 52.4% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e / (((1.0d0 / v) + (e / v)) + (e * (v * (-0.3333333333333333d0))))
end function

Java

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

Python

def code(e, v):
	return e / (((1.0 / v) + (e / v)) + (e * (v * -0.3333333333333333)))

Julia

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

MATLAB

function tmp = code(e, v)
	tmp = e / (((1.0 / v) + (e / v)) + (e * (v * -0.3333333333333333)));
end

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(1.0 / v), $MachinePrecision] + N[(e / v), $MachinePrecision]), $MachinePrecision] + N[(e * N[(v * -0.3333333333333333), $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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

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. associate-/l* 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around 0 60.1%

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

8. Taylor expanded in e around inf 59.8%

   \[\leadsto \frac{e}{\color{blue}{-0.3333333333333333 \cdot \left(e \cdot v\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]
9. Step-by-step derivation

   1. *-commutative 59.8%

      \[\leadsto \frac{e}{\color{blue}{\left(e \cdot v\right) \cdot -0.3333333333333333} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

   2. associate-*l* 59.8%

      \[\leadsto \frac{e}{\color{blue}{e \cdot \left(v \cdot -0.3333333333333333\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

10. Simplified 59.8%

    \[\leadsto \frac{e}{\color{blue}{e \cdot \left(v \cdot -0.3333333333333333\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

11. Final simplification 59.8%

    \[\leadsto \frac{e}{\left(\frac{1}{v} + \frac{e}{v}\right) + e \cdot \left(v \cdot -0.3333333333333333\right)} \]

Alternative 7: 51.3% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v - N[(e * 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 59.3%

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

   1. associate-/l* 59.2%

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

   2. associate-/r/ 59.3%

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

   3. +-commutative 59.3%

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

6. Simplified 59.3%

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

7. Taylor expanded in e around 0 59.0%

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

   1. mul-1-neg 59.0%

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

   2. unsub-neg 59.0%

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

   3. unpow2 59.0%

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

9. Simplified 59.0%

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

10. Taylor expanded in e around 0 59.0%

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

    1. +-commutative 59.0%

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

    2. mul-1-neg 59.0%

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

    3. distribute-lft-neg-in 59.0%

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

    4. unpow2 59.0%

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

    5. distribute-rgt-neg-out 59.0%

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

    6. associate-*l* 59.0%

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

    7. distribute-lft-out 59.0%

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

    8. distribute-lft-neg-out 59.0%

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

    9. unsub-neg 59.0%

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

12. Simplified 59.0%

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

13. Final simplification 59.0%

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

Alternative 8: 51.8% accurate, 29.9× speedup

Math

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

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(e * Float64(v / Float64(e + 1.0)))
end

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v / 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-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

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. associate-/l* 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around 0 59.3%

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

   1. +-commutative 59.3%

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

   2. associate-*r/ 59.3%

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

9. Simplified 59.3%

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

10. Final simplification 59.3%

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

Alternative 9: 50.8% 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-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 59.3%

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

   1. associate-/l* 59.2%

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

   2. associate-/r/ 59.3%

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

   3. +-commutative 59.3%

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

6. Simplified 59.3%

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

7. Taylor expanded in e around 0 58.5%

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

8. Final simplification 58.5%

   \[\leadsto e \cdot v \]

Alternative 10: 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-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 59.3%

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

   1. associate-/l* 59.2%

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

   2. associate-/r/ 59.3%

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

   3. +-commutative 59.3%

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

6. Simplified 59.3%

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

7. Taylor expanded in e around inf 4.8%

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

8. Final simplification 4.8%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023283 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))