Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

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

\[\begin{array}{l} \\ \sin v \cdot \left(e \cdot \frac{1}{\mathsf{fma}\left(e, \cos v, 1\right)}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-udef 99.7%

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

   4. associate-*l/ 99.7%

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

   5. div-inv 99.7%

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

   6. fma-udef 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-*l* 99.7%

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

   9. +-commutative 99.7%

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

   10. fma-udef 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

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(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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around inf 99.7%

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

6. Final simplification 99.7%

   \[\leadsto e \cdot \frac{\sin v}{1 + e \cdot \cos v} \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(sin(v) * 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[(N[Sin[v], $MachinePrecision] * e), $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. Add Preprocessing

3. Final simplification 99.7%

   \[\leadsto \frac{\sin v \cdot e}{1 + e \cdot \cos v} \]
4. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

5. Final simplification 99.6%

   \[\leadsto \frac{\sin v}{\cos v + \frac{1}{e}} \]
6. Add Preprocessing

Alternative 5: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around inf 99.7%

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

6. Taylor expanded in v around 0 97.9%

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

7. Final simplification 97.9%

   \[\leadsto e \cdot \frac{\sin v}{1 + e} \]
8. Add Preprocessing

Alternative 6: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \frac{\sin v \cdot e}{1 + e} \]
5. Add Preprocessing

Alternative 7: 97.8% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in e around 0 96.9%

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

6. Final simplification 96.9%

   \[\leadsto \sin v \cdot e \]
7. Add Preprocessing

Alternative 8: 52.1% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(N[(1.0 / v), $MachinePrecision] + N[(1.0 / N[(v * e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(v * N[(0.5 + N[(-0.16666666666666666 * N[(1.0 + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $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. *-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. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

   3. +-commutative 99.2%

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

6. Applied egg-rr 99.2%

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

   1. clear-num 98.6%

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

   2. inv-pow 98.6%

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

   3. unpow2 98.6%

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

   4. add-sqr-sqrt 98.9%

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

8. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

10. Simplified 98.9%

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

11. Taylor expanded in v around 0 48.8%

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

12. Final simplification 48.8%

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

Alternative 9: 51.9% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 + e), $MachinePrecision] / v), $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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 48.2%

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

   1. associate-/l* 48.2%

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

   2. +-commutative 48.2%

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

7. Simplified 48.2%

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

8. Final simplification 48.2%

   \[\leadsto \frac{e}{\frac{1 + e}{v}} \]
9. Add Preprocessing

Alternative 10: 52.0% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v * e), $MachinePrecision] / N[(1.0 + e), $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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 48.2%

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

6. Final simplification 48.2%

   \[\leadsto \frac{v \cdot e}{1 + e} \]
7. Add Preprocessing

Alternative 11: 51.0% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 48.2%

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

   1. associate-/l* 48.2%

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

   2. +-commutative 48.2%

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

7. Simplified 48.2%

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

8. Taylor expanded in e around 0 47.2%

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

9. Final simplification 47.2%

   \[\leadsto v \cdot e \]
10. Add Preprocessing

Alternative 12: 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. *-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/ 99.7%

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

   4. +-commutative 99.7%

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

   5. cos-neg 99.7%

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

   6. fma-def 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 48.2%

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

   1. associate-/l* 48.2%

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

   2. +-commutative 48.2%

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

7. Simplified 48.2%

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

8. Taylor expanded in e around inf 4.4%

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

9. Final simplification 4.4%

   \[\leadsto v \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))