Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 12

Speedup: 0.5×

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos v, e, -1\right) \cdot \frac{e \cdot \sin v}{{\left(\cos v \cdot e\right)}^{2} - 1} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (* (fma (cos v) e -1.0) (/ (* e (sin v)) (- (pow (* (cos v) e) 2.0) 1.0))))

C

double code(double e, double v) {
	return fma(cos(v), e, -1.0) * ((e * sin(v)) / (pow((cos(v) * e), 2.0) - 1.0));
}

Julia

function code(e, v)
	return Float64(fma(cos(v), e, -1.0) * Float64(Float64(e * sin(v)) / Float64((Float64(cos(v) * e) ^ 2.0) - 1.0)))
end

Wolfram

code[e_, v_] := N[(N[(N[Cos[v], $MachinePrecision] * e + -1.0), $MachinePrecision] * N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{\sin v \cdot e}{\color{blue}{{\left(e \cdot \cos v\right)}^{2}} - 1} \cdot \left(e \cdot \cos v - 1\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \frac{\sin v \cdot e}{\color{blue}{{\left(e \cdot \cos v\right)}^{2}} - 1} \cdot \left(e \cdot \cos v - 1\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\sin v \cdot e}{{\color{blue}{\left(e \cdot \cos v\right)}}^{2} - 1} \cdot \left(e \cdot \cos v - 1\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{\sin v \cdot e}{{\color{blue}{\left(\cos v \cdot e\right)}}^{2} - 1} \cdot \left(e \cdot \cos v - 1\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\sin v \cdot e}{{\color{blue}{\left(\cos v \cdot e\right)}}^{2} - 1} \cdot \left(e \cdot \cos v - 1\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\sin v \cdot e}{{\left(\cos v \cdot e\right)}^{2} - 1} \cdot \mathsf{fma}\left(\cos v, e, -1\right)} \]

5. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\cos v, e, -1\right) \cdot \frac{e \cdot \sin v}{{\left(\cos v \cdot e\right)}^{2} - 1} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 3: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 98.7

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

5. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{e}{1 + e} \cdot v\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 2.95e-8) (* (/ e (+ 1.0 e)) v) (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 2.95e-8) {
		tmp = (e / (1.0 + e)) * v;
	} 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 <= 2.95d-8) then
        tmp = (e / (1.0d0 + e)) * v
    else
        tmp = e * sin(v)
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 2.95e-8) {
		tmp = (e / (1.0 + e)) * v;
	} else {
		tmp = e * Math.sin(v);
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 2.95e-8:
		tmp = (e / (1.0 + e)) * v
	else:
		tmp = e * math.sin(v)
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 2.95e-8)
		tmp = Float64(Float64(e / Float64(1.0 + e)) * v);
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 2.95e-8)
		tmp = (e / (1.0 + e)) * v;
	else
		tmp = e * sin(v);
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 2.95e-8], N[(N[(e / N[(1.0 + e), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2.9499999999999999e-8

   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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 2.9499999999999999e-8 < v

   1. Initial program 99.4%

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

   3. Taylor expanded in e around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{e}{1 + e} \cdot v\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Final simplification 50.7%

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

Alternative 7: 50.0% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 48.7%

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

9. Taylor expanded in e around 0

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

11. Applied rewrites 49.6%

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

13. Applied rewrites 49.6%

    \[\leadsto \mathsf{fma}\left(v, e, \left(\left(-e\right) \cdot v\right) \cdot e\right) \]
14. Add Preprocessing

Alternative 8: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 45.4%

   \[\leadsto \frac{0 - \frac{e}{-1 - e} \cdot \frac{e}{-1 - e}}{0 + \frac{e}{-1 - e}} \cdot v \]

8. Taylor expanded in e around 0

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

10. Applied rewrites 49.6%

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

Alternative 9: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in e around 0

   \[\leadsto \left(e \cdot \left(1 + -1 \cdot e\right)\right) \cdot v \]
7. Step-by-step derivation

8. Applied rewrites 49.6%

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

Alternative 10: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 48.7%

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

9. Taylor expanded in e around 0

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

11. Applied rewrites 49.6%

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

12. Final simplification 49.6%

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

Alternative 11: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 49.6%

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

Alternative 12: 49.5% accurate, 37.5× 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 99.7%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 48.7%

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

9. Final simplification 48.7%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))