Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.4s

Alternatives: 11

Speedup: N/A×

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

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

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 99.8%

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

   1. *-commutative 99.8%

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

   2. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in v around inf 99.8%

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

6. Final simplification 99.8%

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

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. cos-neg 99.8%

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

   3. associate-/l* 99.5%

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

   4. +-commutative 99.5%

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

   5. cos-neg 99.5%

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

   6. metadata-eval 99.5%

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

   7. sub-neg 99.5%

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

   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 3: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] / N[(1.0 + 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. cos-neg 99.8%

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

   3. associate-/l* 99.5%

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

   4. +-commutative 99.5%

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

   5. cos-neg 99.5%

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

   6. metadata-eval 99.5%

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

   7. sub-neg 99.5%

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

   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. Taylor expanded in v around 0 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 98.9% 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 99.8%

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

3. Taylor expanded in v around 0 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 97.9% 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. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in e around 0 97.8%

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

6. Final simplification 97.8%

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

Alternative 6: 53.2% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(v * N[(N[(e * -0.5), $MachinePrecision] - N[(N[(1.0 + e), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $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. *-commutative 99.8%

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

   2. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing
5. 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. fma-udef 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-/l* 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-udef 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in v around 0 48.9%

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

8. Final simplification 48.9%

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

Alternative 7: 52.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing
5. 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. fma-udef 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-/l* 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-udef 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in e around 0 97.6%

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

8. Taylor expanded in v around 0 48.1%

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

9. Final simplification 48.1%

   \[\leadsto \frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \]
10. Add Preprocessing

Alternative 8: 51.7% accurate, 29.9× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* e (* v (- 1.0 e))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in v around 0 47.6%

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

   1. +-commutative 47.6%

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

7. Simplified 47.6%

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

8. Taylor expanded in e around 0 47.2%

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

   1. mul-1-neg 47.2%

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

   2. distribute-rgt-neg-out 47.2%

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

10. Simplified 47.2%

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

11. Taylor expanded in v around 0 47.2%

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

    1. neg-mul-1 47.2%

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

    2. sub-neg 47.2%

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

13. Simplified 47.2%

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

14. Final simplification 47.2%

    \[\leadsto e \cdot \left(v \cdot \left(1 - e\right)\right) \]
15. Add Preprocessing

Alternative 9: 52.1% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in v around 0 47.6%

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

   1. +-commutative 47.6%

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

7. Simplified 47.6%

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

8. Final simplification 47.6%

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

Alternative 10: 51.1% 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. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in v around 0 47.6%

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

   1. +-commutative 47.6%

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

7. Simplified 47.6%

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

8. Taylor expanded in e around 0 47.0%

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

9. Final simplification 47.0%

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

Alternative 11: 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. cos-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. +-commutative 99.8%

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

   5. cos-neg 99.8%

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

   6. 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. Add Preprocessing

5. Taylor expanded in v around 0 47.6%

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

   1. +-commutative 47.6%

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

7. Simplified 47.6%

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

8. Taylor expanded in e around inf 4.3%

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

9. Final simplification 4.3%

   \[\leadsto v \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))