Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 15

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

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

FPCore

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

C

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

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 = ((e * sin(v)) / ((t_0 ** 2.0d0) + (-1.0d0))) * (t_0 + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $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. associate-/l* 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-udef 99.8%

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

   4. flip-+ 99.8%

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

   5. associate-/r/ 99.8%

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

   6. pow2 99.8%

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

   7. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] / N[(N[(e + 1.0), $MachinePrecision] / e), $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 98.8%

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

   1. +-commutative 98.8%

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

6. Simplified 98.8%

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

7. Final simplification 98.8%

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

Alternative 6: 98.8% 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 99.8%

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

2. Taylor expanded in v around 0 99.0%

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

3. Final simplification 99.0%

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

Alternative 7: 97.6% 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in e around 0 97.8%

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

5. Final simplification 97.8%

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

Alternative 8: 53.0% 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-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 51.6%

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

7. Taylor expanded in e around 0 51.6%

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

8. Final simplification 51.6%

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

Alternative 9: 52.6% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 51.6%

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

7. Taylor expanded in e around inf 50.8%

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

   1. *-commutative 50.8%

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

9. Simplified 50.8%

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

10. Final simplification 50.8%

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

Alternative 10: 51.8% 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-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 51.6%

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

7. Taylor expanded in e around 0 50.4%

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

8. Final simplification 50.4%

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

Alternative 11: 51.9% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 50.2%

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

   1. +-commutative 50.2%

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

8. Simplified 50.2%

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

9. Final simplification 50.2%

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

Alternative 12: 51.9% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around inf 99.6%

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

7. Taylor expanded in v around 0 50.3%

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

   1. *-commutative 50.3%

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

   2. associate-/l* 50.2%

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

   3. *-rgt-identity 50.2%

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

   4. associate-*r/ 50.2%

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

   5. +-commutative 50.2%

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

   6. distribute-lft1-in 50.2%

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

   7. rgt-mult-inverse 50.2%

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

9. Simplified 50.2%

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

10. Final simplification 50.2%

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

Alternative 13: 51.9% 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 99.8%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 50.3%

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

   1. +-commutative 50.3%

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

6. Simplified 50.3%

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

7. Final simplification 50.3%

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

Alternative 14: 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-*r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 50.3%

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

   1. +-commutative 50.3%

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

6. Simplified 50.3%

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

7. Taylor expanded in e around 0 49.1%

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

8. Final simplification 49.1%

   \[\leadsto e \cdot v \]

Alternative 15: 4.6% 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 50.3%

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

   1. +-commutative 50.3%

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

6. Simplified 50.3%

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

7. Taylor expanded in e around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023301 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))