Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

Alternatives: 13

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(e / 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-*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. Step-by-step derivation

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

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. *-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 4: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 97.9%

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

   1. +-commutative 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in e around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. unpow2 95.9%

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

9. Simplified 95.9%

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

10. Final simplification 95.9%

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

Alternative 5: 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 97.9%

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

   1. +-commutative 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 6: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;e \cdot \sin v\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{e \cdot \left(v \cdot -0.3333333333333333\right) + \left(\frac{1}{v} + \frac{e}{v}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= e 1.5e-7)
   (* e (sin v))
   (/ e (+ (* e (* v -0.3333333333333333)) (+ (/ 1.0 v) (/ e v))))))

C

double code(double e, double v) {
	double tmp;
	if (e <= 1.5e-7) {
		tmp = e * sin(v);
	} else {
		tmp = e / ((e * (v * -0.3333333333333333)) + ((1.0 / v) + (e / v)));
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (e <= 1.5d-7) then
        tmp = e * sin(v)
    else
        tmp = e / ((e * (v * (-0.3333333333333333d0))) + ((1.0d0 / v) + (e / v)))
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (e <= 1.5e-7) {
		tmp = e * Math.sin(v);
	} else {
		tmp = e / ((e * (v * -0.3333333333333333)) + ((1.0 / v) + (e / v)));
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if e <= 1.5e-7:
		tmp = e * math.sin(v)
	else:
		tmp = e / ((e * (v * -0.3333333333333333)) + ((1.0 / v) + (e / v)))
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (e <= 1.5e-7)
		tmp = Float64(e * sin(v));
	else
		tmp = Float64(e / Float64(Float64(e * Float64(v * -0.3333333333333333)) + Float64(Float64(1.0 / v) + Float64(e / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (e <= 1.5e-7)
		tmp = e * sin(v);
	else
		tmp = e / ((e * (v * -0.3333333333333333)) + ((1.0 / v) + (e / v)));
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[e, 1.5e-7], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision], N[(e / N[(N[(e * N[(v * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / v), $MachinePrecision] + N[(e / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if e < 1.4999999999999999e-7

   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 98.7%

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

   if 1.4999999999999999e-7 < e

   1. Initial program 99.2%

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

      1. associate-*l/ 99.1%

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

      2. +-commutative 99.1%

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

      3. fma-def 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in v around inf 99.2%

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

      1. associate-/l* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in v around 0 67.1%

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

   8. Taylor expanded in e around inf 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*l* 67.6%

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

   10. Simplified 67.6%

       \[\leadsto \frac{e}{\color{blue}{e \cdot \left(v \cdot -0.3333333333333333\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;e \cdot \sin v\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{e \cdot \left(v \cdot -0.3333333333333333\right) + \left(\frac{1}{v} + \frac{e}{v}\right)}\\ \end{array} \]

Alternative 7: 53.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\left(\frac{1}{v} + \frac{e}{v}\right) + v \cdot \left(0.16666666666666666 + e \cdot -0.3333333333333333\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (+
   (+ (/ 1.0 v) (/ e v))
   (* v (+ 0.16666666666666666 (* e -0.3333333333333333))))))

C

double code(double e, double v) {
	return e / (((1.0 / v) + (e / v)) + (v * (0.16666666666666666 + (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 * (0.16666666666666666d0 + (e * (-0.3333333333333333d0)))))
end function

Java

public static double code(double e, double v) {
	return e / (((1.0 / v) + (e / v)) + (v * (0.16666666666666666 + (e * -0.3333333333333333))));
}

Python

def code(e, v):
	return e / (((1.0 / v) + (e / v)) + (v * (0.16666666666666666 + (e * -0.3333333333333333))))

Julia

function code(e, v)
	return Float64(e / Float64(Float64(Float64(1.0 / v) + Float64(e / v)) + Float64(v * Float64(0.16666666666666666 + Float64(e * -0.3333333333333333)))))
end

MATLAB

function tmp = code(e, v)
	tmp = e / (((1.0 / v) + (e / v)) + (v * (0.16666666666666666 + (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[(0.16666666666666666 + N[(e * -0.3333333333333333), $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-*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-def 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 50.2%

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

8. Taylor expanded in e around 0 50.2%

   \[\leadsto \frac{e}{v \cdot \color{blue}{\left(0.16666666666666666 + -0.3333333333333333 \cdot e\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]
9. Step-by-step derivation

   1. *-commutative 50.2%

      \[\leadsto \frac{e}{v \cdot \left(0.16666666666666666 + \color{blue}{e \cdot -0.3333333333333333}\right) + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

10. Simplified 50.2%

    \[\leadsto \frac{e}{v \cdot \color{blue}{\left(0.16666666666666666 + e \cdot -0.3333333333333333\right)} + \left(\frac{1}{v} + \frac{e}{v}\right)} \]

11. Final simplification 50.2%

    \[\leadsto \frac{e}{\left(\frac{1}{v} + \frac{e}{v}\right) + v \cdot \left(0.16666666666666666 + e \cdot -0.3333333333333333\right)} \]

Alternative 8: 53.4% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(e * N[(v * -0.3333333333333333), $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-def 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 50.2%

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

8. Taylor expanded in e around inf 50.0%

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

   1. *-commutative 50.0%

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

   2. associate-*l* 50.0%

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

10. Simplified 50.0%

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

11. Final simplification 50.0%

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

Alternative 9: 52.3% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 48.9%

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

   1. associate-/l* 48.9%

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

   2. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around 0 47.0%

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

   1. +-commutative 47.0%

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

   2. associate-*r* 47.0%

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

   3. distribute-rgt-out 47.0%

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

   4. mul-1-neg 47.0%

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

   5. unsub-neg 47.0%

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

   6. unpow2 47.0%

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

9. Simplified 47.0%

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

10. Final simplification 47.0%

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

Alternative 10: 52.6% 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-*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 48.9%

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

   1. associate-/l* 48.9%

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

   2. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Final simplification 48.9%

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

Alternative 11: 52.7% 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-*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 48.9%

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

5. Final simplification 48.9%

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

Alternative 12: 51.8% 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 48.9%

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

   1. associate-/l* 48.9%

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

   2. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around 0 46.0%

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

8. Final simplification 46.0%

   \[\leadsto e \cdot v \]

Alternative 13: 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 48.9%

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

   1. associate-/l* 48.9%

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

   2. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around inf 4.6%

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

8. Final simplification 4.6%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023272 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))