Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. associate-*l* 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return sin(v) * (1.0 / (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) * (1.0d0 / (cos(v) + (1.0d0 / e)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(1.0 / N[(N[Cos[v], $MachinePrecision] + N[(1.0 / e), $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* 99.6%

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

   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. clear-num 98.8%

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

   2. associate-/r/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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* 99.6%

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

   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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. associate-*l* 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in v around 0 99.0%

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

   1. +-commutative 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 6: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-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* 99.6%

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

   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. Taylor expanded in e around 0 98.2%

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

6. Final simplification 98.2%

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

Alternative 7: 52.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. associate-*l* 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. log1p-expm1-u 99.7%

      \[\leadsto e \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin v \cdot \frac{1}{\mathsf{fma}\left(e, \cos v, 1\right)}\right)\right)} \]

   2. un-div-inv 99.7%

      \[\leadsto e \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}}\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto e \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. log1p-expm1-u 99.8%

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

   2. clear-num 99.6%

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

   3. frac-2neg 99.6%

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

   4. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. distribute-neg-frac 99.6%

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

   2. neg-sub0 99.6%

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

   3. fma-udef 99.6%

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

   4. +-commutative 99.6%

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

   5. associate--r+ 99.6%

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

   6. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in v around 0 54.1%

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

12. Final simplification 54.1%

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

Alternative 8: 51.3% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * 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. Add Preprocessing

3. Taylor expanded in v around 0 52.8%

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

   1. associate-/l* 52.7%

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

   2. associate-/r/ 52.8%

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

5. Simplified 52.8%

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

6. Final simplification 52.8%

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

Alternative 9: 50.3% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0 52.8%

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

   1. associate-/l* 52.7%

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

   2. associate-/r/ 52.8%

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

5. Simplified 52.8%

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

6. Taylor expanded in e around 0 51.9%

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

7. Final simplification 51.9%

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

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. Add Preprocessing

3. Taylor expanded in v around 0 52.8%

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

   1. associate-/l* 52.7%

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

   2. associate-/r/ 52.8%

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

5. Simplified 52.8%

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

6. Taylor expanded in e around inf 4.6%

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

7. Final simplification 4.6%

   \[\leadsto v \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))