Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

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

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

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(e * Float64(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[(e * N[(N[Sin[v], $MachinePrecision] / N[(N[(e * N[Cos[v], $MachinePrecision]), $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. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. remove-double-neg 99.8%

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

   2. neg-sub0 99.8%

      \[\leadsto \frac{\sin v}{\color{blue}{0 - \left(-\mathsf{fma}\left(e, \cos v, 1\right)\right)}} \cdot e \]

   3. fma-udef 99.8%

      \[\leadsto \frac{\sin v}{0 - \left(-\color{blue}{\left(e \cdot \cos v + 1\right)}\right)} \cdot e \]

   4. distribute-neg-in 99.8%

      \[\leadsto \frac{\sin v}{0 - \color{blue}{\left(\left(-e \cdot \cos v\right) + \left(-1\right)\right)}} \cdot e \]

   5. metadata-eval 99.8%

      \[\leadsto \frac{\sin v}{0 - \left(\left(-e \cdot \cos v\right) + \color{blue}{-1}\right)} \cdot e \]

   6. associate--r+ 99.8%

      \[\leadsto \frac{\sin v}{\color{blue}{\left(0 - \left(-e \cdot \cos v\right)\right) - -1}} \cdot e \]

   7. neg-sub0 99.8%

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

   8. remove-double-neg 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

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* 99.6%

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

   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 3: 98.7% 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. Taylor expanded in v around 0 99.3%

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

   1. associate-/l* 99.1%

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

   2. remove-double-neg 99.1%

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

   3. neg-sub0 99.1%

      \[\leadsto \frac{\color{blue}{0 - \left(-e\right)}}{\frac{1 + e}{\sin v}} \]

   4. div-sub 99.1%

      \[\leadsto \color{blue}{\frac{0}{\frac{1 + e}{\sin v}} - \frac{-e}{\frac{1 + e}{\sin v}}} \]

   5. +-commutative 99.1%

      \[\leadsto \frac{0}{\frac{\color{blue}{e + 1}}{\sin v}} - \frac{-e}{\frac{1 + e}{\sin v}} \]

   6. +-commutative 99.1%

      \[\leadsto \frac{0}{\frac{e + 1}{\sin v}} - \frac{-e}{\frac{\color{blue}{e + 1}}{\sin v}} \]

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{0}{\frac{e + 1}{\sin v}} - \frac{-e}{\frac{e + 1}{\sin v}}} \]
5. Step-by-step derivation

   1. div0 99.1%

      \[\leadsto \color{blue}{0} - \frac{-e}{\frac{e + 1}{\sin v}} \]

   2. sub0-neg 99.1%

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

   3. distribute-frac-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. associate-/l* 99.3%

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

   6. remove-double-neg 99.3%

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

   7. associate-*l/ 99.3%

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

   8. +-commutative 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 98.7% accurate, 2.0× speedup

Math

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

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(Float64(sin(v) * 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[(N[Sin[v], $MachinePrecision] * e), $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 99.3%

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

3. Final simplification 99.3%

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

Alternative 5: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in e around 0 98.4%

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

5. Final simplification 98.4%

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

Alternative 6: 52.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = -e / ((((-1.0d0) - e) / v) - (v * ((e * (-0.5d0)) + ((e + 1.0d0) * 0.16666666666666666d0))))
end function

Java

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

Python

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

Julia

function code(e, v)
	return Float64(Float64(-e) / Float64(Float64(Float64(-1.0 - e) / v) - Float64(v * Float64(Float64(e * -0.5) + Float64(Float64(e + 1.0) * 0.16666666666666666)))))
end

MATLAB

function tmp = code(e, v)
	tmp = -e / (((-1.0 - e) / v) - (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666))));
end

Wolfram

code[e_, v_] := N[((-e) / N[(N[(N[(-1.0 - e), $MachinePrecision] / v), $MachinePrecision] - N[(v * N[(N[(e * -0.5), $MachinePrecision] + N[(N[(e + 1.0), $MachinePrecision] * 0.16666666666666666), $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. *-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. associate-/l* 99.6%

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

   4. frac-2neg 99.6%

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

   5. associate-/r/ 99.8%

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

   6. fma-udef 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. metadata-eval 99.8%

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

   10. unsub-neg 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-/l* 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in v around 0 50.4%

   \[\leadsto \frac{-e}{\color{blue}{-1 \cdot \left(v \cdot \left(-0.5 \cdot e - -0.16666666666666666 \cdot \left(1 + e\right)\right)\right) + -1 \cdot \frac{1 + e}{v}}} \]
9. Step-by-step derivation

   1. distribute-lft-out 50.4%

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

   2. cancel-sign-sub-inv 50.4%

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

   3. *-commutative 50.4%

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

   4. metadata-eval 50.4%

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

   5. +-commutative 50.4%

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

   6. +-commutative 50.4%

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

10. Simplified 50.4%

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

11. Final simplification 50.4%

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

Alternative 7: 50.9% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 49.4%

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

   1. +-commutative 49.4%

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

6. Simplified 49.4%

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

7. Taylor expanded in e around 0 48.8%

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

   1. mul-1-neg 48.8%

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

   2. unsub-neg 48.8%

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

9. Simplified 48.8%

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

10. Final simplification 48.8%

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

Alternative 8: 51.4% 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. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 49.4%

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

   1. +-commutative 49.4%

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

6. Simplified 49.4%

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

7. Final simplification 49.4%

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

Alternative 9: 50.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 49.4%

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

   1. +-commutative 49.4%

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

6. Simplified 49.4%

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

7. Taylor expanded in e around 0 48.5%

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

8. Final simplification 48.5%

   \[\leadsto v \cdot e \]

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. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 49.4%

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

   1. +-commutative 49.4%

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

6. Simplified 49.4%

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

7. Taylor expanded in e around inf 4.5%

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

8. Final simplification 4.5%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023297 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))