Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

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/ 99.8%

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

   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. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

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(e * Float64(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[(e * N[(N[Sin[v], $MachinePrecision] * N[(1.0 - N[(e * N[Cos[v], $MachinePrecision]), $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. Taylor expanded in e around 0 98.7%

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

   1. *-lft-identity 98.7%

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

   2. mul-1-neg 98.7%

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

   3. associate-*r* 98.7%

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

   4. distribute-lft-neg-in 98.7%

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

   5. distribute-rgt-in 98.7%

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

   6. sub-neg 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $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. Final simplification 99.6%

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

Alternative 4: 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 5: 98.7% 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 98.5%

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

   1. +-commutative 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 6: 97.8% 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 97.9%

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

5. Final simplification 97.9%

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

Alternative 7: 53.2% 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 49.4%

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

5. Taylor expanded in e around 0 49.4%

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

6. Final simplification 49.4%

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

Alternative 8: 52.7% accurate, 13.9× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return e / (((1.0 / v) + (e / v)) + (v * (e * -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)) + (v * (e * (-0.3333333333333333d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 49.4%

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

5. Taylor expanded in e around inf 48.7%

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

   1. *-commutative 48.7%

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

   2. *-commutative 48.7%

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

   3. associate-*l* 48.7%

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

7. Simplified 48.7%

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

8. Final simplification 48.7%

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

Alternative 9: 52.1% 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 e around 0 97.7%

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

5. Taylor expanded in v around 0 48.6%

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

6. Final simplification 48.6%

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

Alternative 10: 51.5% 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-*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 48.0%

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

   1. +-commutative 48.0%

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

6. Simplified 48.0%

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

7. Taylor expanded in e around 0 47.4%

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

   1. mul-1-neg 47.4%

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

   2. unsub-neg 47.4%

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

9. Simplified 47.4%

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

10. Final simplification 47.4%

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

Alternative 11: 52.1% 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-*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 48.0%

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

   1. +-commutative 48.0%

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

6. Simplified 48.0%

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

7. Final simplification 48.0%

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

Alternative 12: 51.0% 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 48.0%

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

   1. +-commutative 48.0%

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

6. Simplified 48.0%

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

7. Taylor expanded in e around 0 47.2%

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

8. Final simplification 47.2%

   \[\leadsto e \cdot v \]

Reproduce

herbie shell --seed 2023309 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))