Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 9

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 99.8%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

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. Step-by-step derivation

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in e around 0 99.6%

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

   1. *-lft-identity 99.6%

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

   2. mul-1-neg 99.6%

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

   3. associate-*r* 99.6%

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

   4. distribute-lft-neg-in 99.6%

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

   5. distribute-rgt-in 99.6%

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

   6. sub-neg 99.6%

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

   7. associate-*l* 99.6%

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

7. Simplified 99.6%

   \[\leadsto \color{blue}{\left(e \cdot \sin v\right) \cdot \left(1 - e \cdot \cos v\right)} \]
8. Add Preprocessing

Alternative 3: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(e + 1.0), $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 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 98.6% 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. Add Preprocessing

3. Taylor expanded in v around 0 99.5%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

   3. +-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. associate-/r* 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in v around inf 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-*r/ 99.5%

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

   3. remove-double-div 99.4%

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

   4. times-frac 99.3%

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

   5. *-lft-identity 99.3%

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

   6. distribute-rgt-in 99.3%

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

   7. rgt-mult-inverse 99.3%

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

   8. *-lft-identity 99.3%

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

8. Simplified 99.3%

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

Alternative 5: 98.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(e + 1.0), $MachinePrecision] / N[Sin[v], $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 99.5%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

   3. +-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. associate-/r* 98.1%

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

5. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

   2. clear-num 99.2%

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

7. Applied egg-rr 99.2%

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

Alternative 6: 97.7% 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. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in e around 0 99.2%

   \[\leadsto \color{blue}{e \cdot \sin v} \]
6. Add Preprocessing

Alternative 7: 51.6% 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. Step-by-step derivation

   1. expm1-log1p-u 99.7%

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

   2. expm1-undefine 79.0%

      \[\leadsto \frac{e \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin v\right)} - 1\right)}}{1 + e \cdot \cos v} \]

4. Applied egg-rr 79.0%

   \[\leadsto \frac{e \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin v\right)} - 1\right)}}{1 + e \cdot \cos v} \]
5. Step-by-step derivation

   1. expm1-define 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around 0 49.4%

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

   1. +-commutative 49.4%

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

   2. *-commutative 49.4%

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

   3. associate-*r/ 49.4%

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

9. Simplified 49.4%

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

Alternative 8: 51.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 49.4%

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

   1. associate-/l* 49.4%

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

   2. +-commutative 49.4%

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

7. Simplified 49.4%

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

Alternative 9: 50.5% 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. Step-by-step derivation

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 49.4%

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

   1. associate-/l* 49.4%

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

   2. +-commutative 49.4%

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

7. Simplified 49.4%

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

8. Taylor expanded in e around 0 49.2%

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

9. Taylor expanded in e around 0 49.2%

   \[\leadsto \color{blue}{e \cdot v} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024173 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))