Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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 99.9%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto e \cdot \frac{\sin v}{1 + e \cdot \cos v} \]
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 99.9%

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

3. Taylor expanded in v around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in e around 0 98.3%

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

Alternative 5: 51.5% accurate, 29.9× speedup

Math

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

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(Float64(e * v) / Float64(e + 1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in v around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

Alternative 6: 51.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0 52.3%

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

   1. +-commutative 52.3%

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

9. Simplified 52.3%

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

Alternative 7: 51.0% 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in v around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in e around 0 52.2%

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

   1. +-commutative 52.2%

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

   2. associate-*r* 52.2%

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

   3. neg-mul-1 52.2%

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

   4. *-lft-identity 52.2%

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

   5. distribute-rgt-out 52.2%

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

   6. metadata-eval 52.2%

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

   7. distribute-neg-in 52.2%

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

   8. +-commutative 52.2%

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

   9. distribute-neg-in 52.2%

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

   10. metadata-eval 52.2%

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

   11. sub-neg 52.2%

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

10. Simplified 52.2%

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

Alternative 8: 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in v around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in e around 0 51.9%

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

Alternative 9: 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. remove-double-neg 99.9%

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

   3. cos-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sin-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sin-neg 99.9%

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

   8. remove-double-neg 99.9%

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

   9. +-commutative 99.9%

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

   10. cos-neg 99.9%

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

   11. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in v around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in e around inf 4.5%

   \[\leadsto \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))