Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 11

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. Final simplification 99.8%

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

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* 99.6%

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

   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. Final simplification 99.6%

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

Alternative 3: 98.5% 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. Taylor expanded in v around 0 98.3%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 31.4%

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

   3. *-commutative 31.4%

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

   4. *-un-lft-identity 31.4%

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

   5. times-frac 31.4%

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

   6. /-rgt-identity 31.4%

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

   7. +-commutative 31.4%

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

4. Applied egg-rr 31.4%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-/r/ 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

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. 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. Taylor expanded in v around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 98.7% 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. Taylor expanded in v around 0 98.3%

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

3. Final simplification 98.3%

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

Alternative 6: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{e \cdot v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 1.5e-25) (/ (* e v) (+ e 1.0)) (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 1.5e-25) {
		tmp = (e * v) / (e + 1.0);
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1.5d-25) then
        tmp = (e * v) / (e + 1.0d0)
    else
        tmp = e * sin(v)
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 1.5e-25) {
		tmp = (e * v) / (e + 1.0);
	} else {
		tmp = e * Math.sin(v);
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 1.5e-25:
		tmp = (e * v) / (e + 1.0)
	else:
		tmp = e * math.sin(v)
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 1.5e-25)
		tmp = Float64(Float64(e * v) / Float64(e + 1.0));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 1.5e-25)
		tmp = (e * v) / (e + 1.0);
	else
		tmp = e * sin(v);
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 1.5e-25], N[(N[(e * v), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.4999999999999999e-25

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.0%

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

   3. Taylor expanded in v around 0 67.6%

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

   if 1.4999999999999999e-25 < v

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. cos-neg 99.5%

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

      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 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/l* 99.4%

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

      11. *-inverses 99.4%

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

      12. /-rgt-identity 99.4%

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

      13. metadata-eval 99.4%

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

      14. associate-/r* 99.4%

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

      15. neg-mul-1 99.4%

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

      16. unsub-neg 99.4%

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

      17. neg-mul-1 99.4%

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

      18. associate-/r* 99.4%

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

      19. metadata-eval 99.4%

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

      20. distribute-neg-frac 99.4%

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

      21. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in e around 0 96.0%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{e \cdot v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]

Alternative 7: 51.2% 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 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. 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. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

   4. +-commutative 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in v around 0 51.5%

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

   1. associate-/l* 51.4%

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

   2. +-commutative 51.4%

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

6. Simplified 51.4%

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

7. Final simplification 51.4%

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

Alternative 8: 51.2% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v / 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* 99.6%

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

   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. Taylor expanded in v around 0 51.5%

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

5. Final simplification 51.5%

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

Alternative 9: 51.3% 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 99.8%

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

2. Taylor expanded in v around 0 98.3%

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

3. Taylor expanded in v around 0 51.5%

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

4. Final simplification 51.5%

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

Alternative 10: 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. 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. Taylor expanded in v around 0 51.5%

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

5. Taylor expanded in e around 0 49.9%

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

6. Final simplification 49.9%

   \[\leadsto e \cdot v \]

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* 99.6%

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

   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. Taylor expanded in v around 0 51.5%

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

5. Taylor expanded in e around inf 4.7%

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

6. Final simplification 4.7%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023331 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))