Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 8

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 99.7%

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

2. Final simplification 99.7%

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

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 99.7%

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

   1. cos-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. cos-neg 99.7%

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

   5. fma-def 99.7%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

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 99.7%

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

   1. *-commutative 99.7%

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

   2. cos-neg 99.7%

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

   3. associate-/l* 99.6%

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

   4. +-commutative 99.6%

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

   5. cos-neg 99.6%

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

   6. metadata-eval 99.6%

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

   7. sub-neg 99.6%

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

   8. div-sub 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-/l* 99.6%

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

   11. *-inverses 99.6%

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

   12. /-rgt-identity 99.6%

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

   13. metadata-eval 99.6%

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

   14. associate-/r* 99.6%

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

   15. neg-mul-1 99.6%

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

   16. unsub-neg 99.6%

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

   17. neg-mul-1 99.6%

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

   18. associate-/r* 99.6%

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

   19. metadata-eval 99.6%

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

   20. distribute-neg-frac 99.6%

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

   21. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;e \cdot \frac{-1}{\frac{-1 - e}{v}}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 1.4e-29) (* e (/ -1.0 (/ (- -1.0 e) v))) (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 1.4e-29) {
		tmp = e * (-1.0 / ((-1.0 - e) / v));
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1.4d-29) then
        tmp = e * ((-1.0d0) / (((-1.0d0) - e) / v))
    else
        tmp = e * sin(v)
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 1.4e-29) {
		tmp = e * (-1.0 / ((-1.0 - e) / v));
	} else {
		tmp = e * Math.sin(v);
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 1.4e-29:
		tmp = e * (-1.0 / ((-1.0 - e) / v))
	else:
		tmp = e * math.sin(v)
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 1.4e-29)
		tmp = Float64(e * Float64(-1.0 / Float64(Float64(-1.0 - e) / v)));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 1.4e-29)
		tmp = e * (-1.0 / ((-1.0 - e) / v));
	else
		tmp = e * sin(v);
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 1.4e-29], N[(e * N[(-1.0 / N[(N[(-1.0 - e), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.4000000000000001e-29

   1. Initial program 99.7%

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

      1. cos-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. +-commutative 99.7%

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

      4. cos-neg 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 60.2%

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

      1. associate-/l* 60.2%

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

      2. +-commutative 60.2%

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

   6. Simplified 60.2%

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

      1. frac-2neg 60.2%

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

      2. div-inv 60.2%

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

      3. distribute-neg-frac 60.2%

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

      4. +-commutative 60.2%

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

      5. distribute-neg-in 60.2%

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

      6. metadata-eval 60.2%

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

   8. Applied egg-rr 60.2%

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

      1. unsub-neg 60.2%

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

   10. Simplified 60.2%

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

   if 1.4000000000000001e-29 < v

   1. Initial program 99.6%

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

      1. cos-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. +-commutative 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in e around 0 99.4%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;e \cdot \frac{-1}{\frac{-1 - e}{v}}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]

Alternative 5: 52.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. cos-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. cos-neg 99.7%

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

   5. fma-def 99.7%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-/r/ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-udef 99.6%

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

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 48.2%

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

7. Final simplification 48.2%

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

Alternative 6: 51.3% 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 99.7%

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

   1. cos-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. cos-neg 99.7%

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

   5. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 47.1%

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

   1. associate-/l* 47.1%

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

   2. +-commutative 47.1%

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

6. Simplified 47.1%

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

   1. clear-num 46.6%

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

   2. associate-/r/ 47.1%

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

   3. clear-num 47.1%

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

8. Applied egg-rr 47.1%

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

9. Final simplification 47.1%

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

Alternative 7: 50.4% 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 99.7%

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

   1. cos-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. cos-neg 99.7%

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

   5. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 47.1%

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

   1. associate-/l* 47.1%

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

   2. +-commutative 47.1%

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

6. Simplified 47.1%

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

7. Taylor expanded in e around 0 44.9%

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

8. Final simplification 44.9%

   \[\leadsto e \cdot v \]

Alternative 8: 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 99.7%

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

   1. cos-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. cos-neg 99.7%

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

   5. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 47.1%

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

   1. associate-/l* 47.1%

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

   2. +-commutative 47.1%

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

6. Simplified 47.1%

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

7. Taylor expanded in e around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023319 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))