Rosa's TurbineBenchmark

Percentage Accurate: 84.0% → 99.8%

Time: 12.5s

Alternatives: 5

Speedup: 1.7×

• 52.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+70}:\\ \;\;\;\;\left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + w \cdot \left(\left(w \cdot r\right) \cdot \left(r \cdot \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+70)
     (+
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* w (* w r)))))
      -1.5)
     (+
      t_0
      (+ -1.5 (* w (* (* w r) (* r (/ (fma v 0.25 -0.375) (- 1.0 v))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+70) {
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	} else {
		tmp = t_0 + (-1.5 + (w * ((w * r) * (r * (fma(v, 0.25, -0.375) / (1.0 - v))))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+70)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(w * r))))) + -1.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(w * Float64(Float64(w * r) * Float64(r * Float64(fma(v, 0.25, -0.375) / Float64(1.0 - v)))))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+70], N[(N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(w * N[(N[(w * r), $MachinePrecision] * N[(r * N[(N[(v * 0.25 + -0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.00000000000000007e70

   1. Initial program 89.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 75.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\sqrt{r \cdot \left(w \cdot w\right)} \cdot \sqrt{r \cdot \left(w \cdot w\right)}\right)}\right)\right) + -1.5 \]

      2. pow2 75.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(\sqrt{r \cdot \left(w \cdot w\right)}\right)}^{2}}\right)\right) + -1.5 \]

      3. *-commutative 75.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\sqrt{\color{blue}{\left(w \cdot w\right) \cdot r}}\right)}^{2}\right)\right) + -1.5 \]

      4. sqrt-prod 50.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right)}}^{2}\right)\right) + -1.5 \]

      5. sqrt-prod 23.5%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

      6. add-sqr-sqrt 51.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{w} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

   5. Applied egg-rr 51.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(w \cdot \sqrt{r}\right)}^{2}}\right)\right) + -1.5 \]
   6. Step-by-step derivation

      1. unpow2 51.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot \sqrt{r}\right) \cdot \left(w \cdot \sqrt{r}\right)\right)}\right)\right) + -1.5 \]

      2. *-commutative 51.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \sqrt{r}\right) \cdot \color{blue}{\left(\sqrt{r} \cdot w\right)}\right)\right)\right) + -1.5 \]

      3. associate-*r* 51.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(\left(w \cdot \sqrt{r}\right) \cdot \sqrt{r}\right) \cdot w\right)}\right)\right) + -1.5 \]

      4. associate-*r* 51.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)\right)} \cdot w\right)\right)\right) + -1.5 \]

      5. add-sqr-sqrt 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \color{blue}{r}\right) \cdot w\right)\right)\right) + -1.5 \]

      6. *-commutative 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot w\right)\right)\right) + -1.5 \]

   7. Applied egg-rr 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)\right) + -1.5 \]

   if 1.00000000000000007e70 < (*.f64 w w)

   1. Initial program 74.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{r \cdot \left(v \cdot 0.25 + -0.375\right)}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right)} \]
   4. Step-by-step derivation

      1. associate-/r* 76.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{r \cdot \left(v \cdot 0.25 + -0.375\right)}{\color{blue}{\frac{1 - v}{r \cdot \left(w \cdot w\right)}}} + -1.5\right) \]

      2. associate-/r/ 76.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{r \cdot \left(v \cdot 0.25 + -0.375\right)}{1 - v} \cdot \left(r \cdot \left(w \cdot w\right)\right)} + -1.5\right) \]

      3. *-commutative 76.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(v \cdot 0.25 + -0.375\right) \cdot r}}{1 - v} \cdot \left(r \cdot \left(w \cdot w\right)\right) + -1.5\right) \]

      4. associate-*l/ 76.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(\frac{v \cdot 0.25 + -0.375}{1 - v} \cdot r\right)} \cdot \left(r \cdot \left(w \cdot w\right)\right) + -1.5\right) \]

      5. +-commutative 76.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(\frac{\color{blue}{-0.375 + v \cdot 0.25}}{1 - v} \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right) + -1.5\right) \]

      6. associate-*r* 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(\frac{-0.375 + v \cdot 0.25}{1 - v} \cdot r\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} + -1.5\right) \]

      7. associate-*r* 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(\left(\frac{-0.375 + v \cdot 0.25}{1 - v} \cdot r\right) \cdot \left(r \cdot w\right)\right) \cdot w} + -1.5\right) \]

      8. +-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(\left(\frac{\color{blue}{v \cdot 0.25 + -0.375}}{1 - v} \cdot r\right) \cdot \left(r \cdot w\right)\right) \cdot w + -1.5\right) \]

      9. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(\color{blue}{\left(r \cdot \frac{v \cdot 0.25 + -0.375}{1 - v}\right)} \cdot \left(r \cdot w\right)\right) \cdot w + -1.5\right) \]

      10. fma-def 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\left(r \cdot \frac{\color{blue}{\mathsf{fma}\left(v, 0.25, -0.375\right)}}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w + -1.5\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(\left(r \cdot \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+70}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + w \cdot \left(\left(w \cdot r\right) \cdot \left(r \cdot \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}\right)\right)\right)\\ \end{array} \]

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-30}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-30)
   (+ -1.5 (+ (* 2.0 (pow r -2.0)) (* -0.25 (* (* w r) (* w r)))))
   (+
    (+
     (/ 2.0 (* r r))
     (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* w (* w r)))))
    -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-30) {
		tmp = -1.5 + ((2.0 * pow(r, -2.0)) + (-0.25 * ((w * r) * (w * r))));
	} else {
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-30) then
        tmp = (-1.5d0) + ((2.0d0 * (r ** (-2.0d0))) + ((-0.25d0) * ((w * r) * (w * r))))
    else
        tmp = ((2.0d0 / (r * r)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (w * (w * r))))) + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-30) {
		tmp = -1.5 + ((2.0 * Math.pow(r, -2.0)) + (-0.25 * ((w * r) * (w * r))));
	} else {
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-30:
		tmp = -1.5 + ((2.0 * math.pow(r, -2.0)) + (-0.25 * ((w * r) * (w * r))))
	else:
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-30)
		tmp = Float64(-1.5 + Float64(Float64(2.0 * (r ^ -2.0)) + Float64(-0.25 * Float64(Float64(w * r) * Float64(w * r)))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(w * r))))) + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-30)
		tmp = -1.5 + ((2.0 * (r ^ -2.0)) + (-0.25 * ((w * r) * (w * r))));
	else
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-30], N[(-1.5 + N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2e-30

   1. Initial program 80.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]

   4. Taylor expanded in v around inf 77.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. unpow2 77.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left({r}^{2} \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) + -1.5 \]

      2. unpow2 77.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right)\right)\right) + -1.5 \]

      3. swap-sqr 96.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

      4. unpow2 96.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}\right) + -1.5 \]

   6. Simplified 96.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot {\left(r \cdot w\right)}^{2}}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 96.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

   8. Applied egg-rr 96.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]
   9. Step-by-step derivation

      1. div-inv 96.4%

         \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

      2. *-commutative 96.4%

         \[\leadsto \left(\color{blue}{\frac{1}{r \cdot r} \cdot 2} + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

      3. pow2 96.4%

         \[\leadsto \left(\frac{1}{\color{blue}{{r}^{2}}} \cdot 2 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

      4. pow-flip 96.5%

         \[\leadsto \left(\color{blue}{{r}^{\left(-2\right)}} \cdot 2 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

      5. metadata-eval 96.5%

         \[\leadsto \left({r}^{\color{blue}{-2}} \cdot 2 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

   10. Applied egg-rr 96.5%

       \[\leadsto \left(\color{blue}{{r}^{-2} \cdot 2} + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]

   if 2e-30 < r

   1. Initial program 90.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 97.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\sqrt{r \cdot \left(w \cdot w\right)} \cdot \sqrt{r \cdot \left(w \cdot w\right)}\right)}\right)\right) + -1.5 \]

      2. pow2 97.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(\sqrt{r \cdot \left(w \cdot w\right)}\right)}^{2}}\right)\right) + -1.5 \]

      3. *-commutative 97.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\sqrt{\color{blue}{\left(w \cdot w\right) \cdot r}}\right)}^{2}\right)\right) + -1.5 \]

      4. sqrt-prod 97.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right)}}^{2}\right)\right) + -1.5 \]

      5. sqrt-prod 50.6%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

      6. add-sqr-sqrt 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{w} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

   5. Applied egg-rr 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(w \cdot \sqrt{r}\right)}^{2}}\right)\right) + -1.5 \]
   6. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot \sqrt{r}\right) \cdot \left(w \cdot \sqrt{r}\right)\right)}\right)\right) + -1.5 \]

      2. *-commutative 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \sqrt{r}\right) \cdot \color{blue}{\left(\sqrt{r} \cdot w\right)}\right)\right)\right) + -1.5 \]

      3. associate-*r* 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(\left(w \cdot \sqrt{r}\right) \cdot \sqrt{r}\right) \cdot w\right)}\right)\right) + -1.5 \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)\right)} \cdot w\right)\right)\right) + -1.5 \]

      5. add-sqr-sqrt 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \color{blue}{r}\right) \cdot w\right)\right)\right) + -1.5 \]

      6. *-commutative 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot w\right)\right)\right) + -1.5 \]

   7. Applied egg-rr 99.8%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)\right) + -1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-30}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 5e+169)
     (+
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* w (* w r)))))
      -1.5)
     (+ -1.5 (+ t_0 (* -0.25 (* (* w r) (* w r))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 5e+169) {
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	} else {
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (w <= 5d+169) then
        tmp = (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (w * (w * r))))) + (-1.5d0)
    else
        tmp = (-1.5d0) + (t_0 + ((-0.25d0) * ((w * r) * (w * r))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 5e+169) {
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	} else {
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 5e+169:
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5
	else:
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 5e+169)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(w * r))))) + -1.5);
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.25 * Float64(Float64(w * r) * Float64(w * r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (w <= 5e+169)
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (w * r))))) + -1.5;
	else
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 5e+169], N[(N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 5.00000000000000017e169

   1. Initial program 86.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 90.6%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 62.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\sqrt{r \cdot \left(w \cdot w\right)} \cdot \sqrt{r \cdot \left(w \cdot w\right)}\right)}\right)\right) + -1.5 \]

      2. pow2 62.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(\sqrt{r \cdot \left(w \cdot w\right)}\right)}^{2}}\right)\right) + -1.5 \]

      3. *-commutative 62.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\sqrt{\color{blue}{\left(w \cdot w\right) \cdot r}}\right)}^{2}\right)\right) + -1.5 \]

      4. sqrt-prod 46.1%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right)}}^{2}\right)\right) + -1.5 \]

      5. sqrt-prod 20.5%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

      6. add-sqr-sqrt 50.3%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot {\left(\color{blue}{w} \cdot \sqrt{r}\right)}^{2}\right)\right) + -1.5 \]

   5. Applied egg-rr 50.3%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{{\left(w \cdot \sqrt{r}\right)}^{2}}\right)\right) + -1.5 \]
   6. Step-by-step derivation

      1. unpow2 50.3%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot \sqrt{r}\right) \cdot \left(w \cdot \sqrt{r}\right)\right)}\right)\right) + -1.5 \]

      2. *-commutative 50.3%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \sqrt{r}\right) \cdot \color{blue}{\left(\sqrt{r} \cdot w\right)}\right)\right)\right) + -1.5 \]

      3. associate-*r* 50.3%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(\left(w \cdot \sqrt{r}\right) \cdot \sqrt{r}\right) \cdot w\right)}\right)\right) + -1.5 \]

      4. associate-*r* 50.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)\right)} \cdot w\right)\right)\right) + -1.5 \]

      5. add-sqr-sqrt 98.1%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\left(w \cdot \color{blue}{r}\right) \cdot w\right)\right)\right) + -1.5 \]

      6. *-commutative 98.1%

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot w\right)\right)\right) + -1.5 \]

   7. Applied egg-rr 98.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)\right) + -1.5 \]

   if 5.00000000000000017e169 < w

   1. Initial program 54.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]

   4. Taylor expanded in v around inf 54.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. unpow2 54.1%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left({r}^{2} \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) + -1.5 \]

      2. unpow2 54.1%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right)\right)\right) + -1.5 \]

      3. swap-sqr 99.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

      4. unpow2 99.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}\right) + -1.5 \]

   6. Simplified 99.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot {\left(r \cdot w\right)}^{2}}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 99.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

   8. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.4 \cdot 10^{+22} \lor \neg \left(v \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;-1.5 + \left(t_0 + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(w \cdot r\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.4e+22) (not (<= v 1.65e-62)))
     (+ -1.5 (+ t_0 (* -0.25 (* (* w r) (* w r)))))
     (+ t_0 (+ -1.5 (* (* w r) (* r (* w -0.375))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.4e+22) || !(v <= 1.65e-62)) {
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	} else {
		tmp = t_0 + (-1.5 + ((w * r) * (r * (w * -0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.4d+22)) .or. (.not. (v <= 1.65d-62))) then
        tmp = (-1.5d0) + (t_0 + ((-0.25d0) * ((w * r) * (w * r))))
    else
        tmp = t_0 + ((-1.5d0) + ((w * r) * (r * (w * (-0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.4e+22) || !(v <= 1.65e-62)) {
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	} else {
		tmp = t_0 + (-1.5 + ((w * r) * (r * (w * -0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.4e+22) or not (v <= 1.65e-62):
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))))
	else:
		tmp = t_0 + (-1.5 + ((w * r) * (r * (w * -0.375))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.4e+22) || !(v <= 1.65e-62))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.25 * Float64(Float64(w * r) * Float64(w * r)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(w * r) * Float64(r * Float64(w * -0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.4e+22) || ~((v <= 1.65e-62)))
		tmp = -1.5 + (t_0 + (-0.25 * ((w * r) * (w * r))));
	else
		tmp = t_0 + (-1.5 + ((w * r) * (r * (w * -0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.4e+22], N[Not[LessEqual[v, 1.65e-62]], $MachinePrecision]], N[(-1.5 + N[(t$95$0 + N[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(w * r), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.4e22 or 1.65000000000000002e-62 < v

   1. Initial program 79.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]

   4. Taylor expanded in v around inf 77.8%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. unpow2 77.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left({r}^{2} \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) + -1.5 \]

      2. unpow2 77.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right)\right)\right) + -1.5 \]

      3. swap-sqr 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

      4. unpow2 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}\right) + -1.5 \]

   6. Simplified 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot {\left(r \cdot w\right)}^{2}}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

   8. Applied egg-rr 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

   if -1.4e22 < v < 1.65000000000000002e-62

   1. Initial program 87.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{r \cdot \left(v \cdot 0.25 + -0.375\right)}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right)} \]

   4. Taylor expanded in v around 0 87.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{-0.375 \cdot r}}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right) \]
   5. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{r \cdot -0.375}}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right) \]

   6. Simplified 87.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{r \cdot -0.375}}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{1 \cdot \left(r \cdot -0.375\right)}}{\frac{\frac{1 - v}{r}}{w \cdot w}} + -1.5\right) \]

      2. div-inv 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{1 \cdot \left(r \cdot -0.375\right)}{\frac{\color{blue}{\left(1 - v\right) \cdot \frac{1}{r}}}{w \cdot w}} + -1.5\right) \]

      3. frac-times 98.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{1 \cdot \left(r \cdot -0.375\right)}{\color{blue}{\frac{1 - v}{w} \cdot \frac{\frac{1}{r}}{w}}} + -1.5\right) \]

      4. *-commutative 98.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(\frac{1 \cdot \left(r \cdot -0.375\right)}{\color{blue}{\frac{\frac{1}{r}}{w} \cdot \frac{1 - v}{w}}} + -1.5\right) \]

      5. times-frac 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{1}{\frac{\frac{1}{r}}{w}} \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}}} + -1.5\right) \]

      6. clear-num 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{w}{\frac{1}{r}}} \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}} + -1.5\right) \]

      7. div-inv 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot \frac{1}{\frac{1}{r}}\right)} \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}} + -1.5\right) \]

      8. clear-num 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot \color{blue}{\frac{r}{1}}\right) \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}} + -1.5\right) \]

      9. /-rgt-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot \color{blue}{r}\right) \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}} + -1.5\right) \]

      10. *-commutative 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot w\right)} \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}} + -1.5\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot w\right) \cdot \frac{r \cdot -0.375}{\frac{1 - v}{w}}} + -1.5\right) \]

   9. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-0.375 \cdot \left(r \cdot w\right)\right)} + -1.5\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(-0.375 \cdot \color{blue}{\left(w \cdot r\right)}\right) + -1.5\right) \]

       2. associate-*l* 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(-0.375 \cdot w\right) \cdot r\right)} + -1.5\right) \]

       3. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot \left(-0.375 \cdot w\right)\right)} + -1.5\right) \]

       4. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot -0.375\right)}\right) + -1.5\right) \]

   11. Simplified 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot -0.375\right)\right)} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.4 \cdot 10^{+22} \lor \neg \left(v \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot r\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 5: 92.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ -1.5 (+ (/ 2.0 (* r r)) (* -0.25 (* (* w r) (* w r))))))

C

double code(double v, double w, double r) {
	return -1.5 + ((2.0 / (r * r)) + (-0.25 * ((w * r) * (w * r))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.25d0) * ((w * r) * (w * r))))
end function

Java

public static double code(double v, double w, double r) {
	return -1.5 + ((2.0 / (r * r)) + (-0.25 * ((w * r) * (w * r))));
}

Python

def code(v, w, r):
	return -1.5 + ((2.0 / (r * r)) + (-0.25 * ((w * r) * (w * r))))

Julia

function code(v, w, r)
	return Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(-0.25 * Float64(Float64(w * r) * Float64(w * r)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5 + ((2.0 / (r * r)) + (-0.25 * ((w * r) * (w * r))));
end

Wolfram

code[v_, w_, r_] := N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 86.6%

   \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5} \]

4. Taylor expanded in v around inf 79.0%

   \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
5. Step-by-step derivation

   1. unpow2 79.0%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left({r}^{2} \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) + -1.5 \]

   2. unpow2 79.0%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right)\right)\right) + -1.5 \]

   3. swap-sqr 96.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

   4. unpow2 96.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}\right) + -1.5 \]

6. Simplified 96.1%

   \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot {\left(r \cdot w\right)}^{2}}\right) + -1.5 \]
7. Step-by-step derivation

   1. unpow2 96.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

8. Applied egg-rr 96.1%

   \[\leadsto \left(\frac{2}{r \cdot r} + -0.25 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right) + -1.5 \]

9. Final simplification 96.1%

   \[\leadsto -1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \]

Reproduce

herbie shell --seed 2023301 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))