Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 98.8%

Time: 13.4s

Alternatives: 9

Speedup: 1.7×

• 53.1% 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: 84.9% 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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{\frac{2}{r_m}}{r_m}\\ \mathbf{if}\;r_m \leq 2.7 \cdot 10^{+205}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{\frac{1 - v}{w}}{r_m \cdot \left(r_m \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r_m \cdot \left(w \cdot \left(r_m \cdot w\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r_m) r_m)))
   (if (<= r_m 2.7e+205)
     (+
      t_0
      (- -1.5 (/ (fma v -0.25 0.375) (/ (/ (- 1.0 v) w) (* r_m (* r_m w))))))
     (+
      t_0
      (-
       -1.5
       (/ (fma v -0.25 0.375) (/ (- 1.0 v) (* r_m (* w (* r_m w))))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (2.0 / r_m) / r_m;
	double tmp;
	if (r_m <= 2.7e+205) {
		tmp = t_0 + (-1.5 - (fma(v, -0.25, 0.375) / (((1.0 - v) / w) / (r_m * (r_m * w)))));
	} else {
		tmp = t_0 + (-1.5 - (fma(v, -0.25, 0.375) / ((1.0 - v) / (r_m * (w * (r_m * w))))));
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(2.0 / r_m) / r_m)
	tmp = 0.0
	if (r_m <= 2.7e+205)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(Float64(1.0 - v) / w) / Float64(r_m * Float64(r_m * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(1.0 - v) / Float64(r_m * Float64(w * Float64(r_m * w)))))));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]}, If[LessEqual[r$95$m, 2.7e+205], N[(t$95$0 + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / w), $MachinePrecision] / N[(r$95$m * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.70000000000000012e205

   1. Initial program 81.1%

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

   3. Simplified 93.6%

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

      1. *-un-lft-identity 93.6%

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

      2. associate-*r* 99.7%

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

      3. times-frac 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/r* 99.0%

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

      2. frac-times 97.5%

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

      3. *-un-lft-identity 97.5%

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

   8. Applied egg-rr 97.5%

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

   if 2.70000000000000012e205 < r

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.7 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{\frac{1 - v}{w}}{r \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{\frac{2}{r_m}}{r_m}\\ \mathbf{if}\;w \leq 1.2 \cdot 10^{+179}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r_m \cdot \left(w \cdot \left(r_m \cdot w\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r_m) r_m)))
   (if (<= w 1.2e+179)
     (+
      t_0
      (- -1.5 (/ (fma v -0.25 0.375) (/ (- 1.0 v) (* r_m (* w (* r_m w)))))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r_m w) (* r_m w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (2.0 / r_m) / r_m;
	double tmp;
	if (w <= 1.2e+179) {
		tmp = t_0 + (-1.5 - (fma(v, -0.25, 0.375) / ((1.0 - v) / (r_m * (w * (r_m * w))))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r_m * w) * (r_m * w))));
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(2.0 / r_m) / r_m)
	tmp = 0.0
	if (w <= 1.2e+179)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(1.0 - v) / Float64(r_m * Float64(w * Float64(r_m * w)))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r_m * w) * Float64(r_m * w)))));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]}, If[LessEqual[w, 1.2e+179], N[(t$95$0 + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 1.20000000000000006e179

   1. Initial program 86.8%

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

   3. Simplified 96.7%

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

   if 1.20000000000000006e179 < w

   1. Initial program 52.9%

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

   3. Simplified 76.5%

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

      1. *-un-lft-identity 76.5%

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

      2. associate-*r* 100.0%

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

      3. times-frac 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. swap-sqr 100.0%

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

      5. unpow2 100.0%

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

      6. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. pow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{\frac{2}{r_m}}{r_m} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1}{r_m \cdot w} \cdot \frac{1 - v}{r_m \cdot w}}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  (/ (/ 2.0 r_m) r_m)
  (-
   -1.5
   (/ (fma v -0.25 0.375) (* (/ 1.0 (* r_m w)) (/ (- 1.0 v) (* r_m w)))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + (-1.5 - (fma(v, -0.25, 0.375) / ((1.0 / (r_m * w)) * ((1.0 - v) / (r_m * w)))));
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(1.0 / Float64(r_m * w)) * Float64(Float64(1.0 - v) / Float64(r_m * w))))))
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(1.0 / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

3. Simplified 94.0%

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

   1. *-un-lft-identity 94.0%

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

   2. associate-*r* 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;v \leq -3.7 \cdot 10^{+68} \lor \neg \left(v \leq 1.05 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\frac{2}{r_m}}{r_m} + \left(-1.5 - \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r_m \cdot r_m} + -0.375 \cdot {\left(r_m \cdot w\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (or (<= v -3.7e+68) (not (<= v 1.05e-30)))
   (+ (/ (/ 2.0 r_m) r_m) (- -1.5 (* (* (* r_m w) (* r_m w)) 0.25)))
   (+ -1.5 (+ (/ 2.0 (* r_m r_m)) (* -0.375 (pow (* r_m w) 2.0))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if ((v <= -3.7e+68) || !(v <= 1.05e-30)) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 - (((r_m * w) * (r_m * w)) * 0.25));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (-0.375 * pow((r_m * w), 2.0)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if ((v <= (-3.7d+68)) .or. (.not. (v <= 1.05d-30))) then
        tmp = ((2.0d0 / r_m) / r_m) + ((-1.5d0) - (((r_m * w) * (r_m * w)) * 0.25d0))
    else
        tmp = (-1.5d0) + ((2.0d0 / (r_m * r_m)) + ((-0.375d0) * ((r_m * w) ** 2.0d0)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if ((v <= -3.7e+68) || !(v <= 1.05e-30)) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 - (((r_m * w) * (r_m * w)) * 0.25));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (-0.375 * Math.pow((r_m * w), 2.0)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if (v <= -3.7e+68) or not (v <= 1.05e-30):
		tmp = ((2.0 / r_m) / r_m) + (-1.5 - (((r_m * w) * (r_m * w)) * 0.25))
	else:
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (-0.375 * math.pow((r_m * w), 2.0)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if ((v <= -3.7e+68) || !(v <= 1.05e-30))
		tmp = Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 - Float64(Float64(Float64(r_m * w) * Float64(r_m * w)) * 0.25)));
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-0.375 * (Float64(r_m * w) ^ 2.0))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if ((v <= -3.7e+68) || ~((v <= 1.05e-30)))
		tmp = ((2.0 / r_m) / r_m) + (-1.5 - (((r_m * w) * (r_m * w)) * 0.25));
	else
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (-0.375 * ((r_m * w) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[Or[LessEqual[v, -3.7e+68], N[Not[LessEqual[v, 1.05e-30]], $MachinePrecision]], N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[Power[N[(r$95$m * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.69999999999999998e68 or 1.0500000000000001e-30 < v

   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 92.9%

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

   5. Taylor expanded in v around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. *-commutative 80.2%

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

      3. unpow2 80.2%

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

      4. unpow2 80.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. pow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -3.69999999999999998e68 < v < 1.0500000000000001e-30

   1. Initial program 84.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in v around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

      6. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.7 \cdot 10^{+68} \lor \neg \left(v \leq 1.05 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\\ t_1 := \frac{\frac{2}{r_m}}{r_m}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+71} \lor \neg \left(v \leq 1.05 \cdot 10^{-30}\right):\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* r_m w) (* r_m w))) (t_1 (/ (/ 2.0 r_m) r_m)))
   (if (or (<= v -2e+71) (not (<= v 1.05e-30)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * (r_m * w);
	double t_1 = (2.0 / r_m) / r_m;
	double tmp;
	if ((v <= -2e+71) || !(v <= 1.05e-30)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r_m * w) * (r_m * w)
    t_1 = (2.0d0 / r_m) / r_m
    if ((v <= (-2d+71)) .or. (.not. (v <= 1.05d-30))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * (r_m * w);
	double t_1 = (2.0 / r_m) / r_m;
	double tmp;
	if ((v <= -2e+71) || !(v <= 1.05e-30)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (r_m * w) * (r_m * w)
	t_1 = (2.0 / r_m) / r_m
	tmp = 0
	if (v <= -2e+71) or not (v <= 1.05e-30):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(r_m * w) * Float64(r_m * w))
	t_1 = Float64(Float64(2.0 / r_m) / r_m)
	tmp = 0.0
	if ((v <= -2e+71) || !(v <= 1.05e-30))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (r_m * w) * (r_m * w);
	t_1 = (2.0 / r_m) / r_m;
	tmp = 0.0;
	if ((v <= -2e+71) || ~((v <= 1.05e-30)))
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]}, If[Or[LessEqual[v, -2e+71], N[Not[LessEqual[v, 1.05e-30]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.0000000000000001e71 or 1.0500000000000001e-30 < v

   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 92.9%

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

   5. Taylor expanded in v around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. *-commutative 80.2%

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

      3. unpow2 80.2%

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

      4. unpow2 80.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. pow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -2.0000000000000001e71 < v < 1.0500000000000001e-30

   1. Initial program 84.3%

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

   3. Simplified 95.2%

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

      1. *-un-lft-identity 95.2%

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

      2. associate-*r* 99.7%

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

      3. times-frac 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

      6. *-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. *-commutative 90.4%

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

       2. pow2 90.4%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+71} \lor \neg \left(v \leq 1.05 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{\frac{2}{r_m}}{r_m} + \left(-1.5 - 0.375 \cdot \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right)\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+ (/ (/ 2.0 r_m) r_m) (- -1.5 (* 0.375 (* (* r_m w) (* r_m w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + (-1.5 - (0.375 * ((r_m * w) * (r_m * w))));
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = ((2.0d0 / r_m) / r_m) + ((-1.5d0) - (0.375d0 * ((r_m * w) * (r_m * w))))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + (-1.5 - (0.375 * ((r_m * w) * (r_m * w))));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((2.0 / r_m) / r_m) + (-1.5 - (0.375 * ((r_m * w) * (r_m * w))))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 - Float64(0.375 * Float64(Float64(r_m * w) * Float64(r_m * w)))))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((2.0 / r_m) / r_m) + (-1.5 - (0.375 * ((r_m * w) * (r_m * w))));
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

3. Simplified 94.0%

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

   1. *-un-lft-identity 94.0%

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

   2. associate-*r* 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0 78.1%

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

   1. *-commutative 78.1%

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

   2. unpow2 78.1%

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

   3. unpow2 78.1%

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

   4. swap-sqr 94.9%

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

   5. unpow2 94.9%

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

   6. *-commutative 94.9%

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

9. Simplified 94.9%

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

    1. *-commutative 95.1%

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

    2. pow2 95.1%

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

11. Applied egg-rr 94.9%

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

12. Final simplification 94.9%

    \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \]
13. Add Preprocessing

Alternative 7: 55.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r_m \leq 0.205:\\ \;\;\;\;\frac{\frac{-2}{r_m}}{-r_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 0.205) (/ (/ -2.0 r_m) (- r_m)) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 0.205) {
		tmp = (-2.0 / r_m) / -r_m;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 0.205d0) then
        tmp = ((-2.0d0) / r_m) / -r_m
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 0.205) {
		tmp = (-2.0 / r_m) / -r_m;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 0.205:
		tmp = (-2.0 / r_m) / -r_m
	else:
		tmp = -1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 0.205)
		tmp = Float64(Float64(-2.0 / r_m) / Float64(-r_m));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 0.205)
		tmp = (-2.0 / r_m) / -r_m;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 0.205], N[(N[(-2.0 / r$95$m), $MachinePrecision] / (-r$95$m)), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.204999999999999988

   1. Initial program 79.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in v around inf 76.5%

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

      1. *-commutative 76.5%

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

      2. *-commutative 76.5%

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

      3. unpow2 76.5%

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

      4. unpow2 76.5%

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

      5. swap-sqr 95.5%

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

      6. unpow2 95.5%

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

      7. *-commutative 95.5%

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

   7. Simplified 95.5%

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

      1. *-commutative 95.5%

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

      2. pow2 95.5%

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

   9. Applied egg-rr 95.5%

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

   10. Taylor expanded in r around 0 58.3%

       \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 58.3%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

       2. associate-/l/ 58.3%

          \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

       3. frac-2neg 58.3%

          \[\leadsto \color{blue}{\frac{-\frac{2}{r}}{-r}} \]

       4. div-inv 58.2%

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

   12. Applied egg-rr 58.2%

       \[\leadsto \color{blue}{\left(-\frac{2}{r}\right) \cdot \frac{1}{-r}} \]
   13. Step-by-step derivation

       1. associate-*r/ 58.3%

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

       2. *-rgt-identity 58.3%

          \[\leadsto \frac{\color{blue}{-\frac{2}{r}}}{-r} \]

       3. distribute-neg-frac 58.3%

          \[\leadsto \frac{\color{blue}{\frac{-2}{r}}}{-r} \]

       4. metadata-eval 58.3%

          \[\leadsto \frac{\frac{\color{blue}{-2}}{r}}{-r} \]

   14. Simplified 58.3%

       \[\leadsto \color{blue}{\frac{\frac{-2}{r}}{-r}} \]

   if 0.204999999999999988 < r

   1. Initial program 93.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around inf 85.8%

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

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. unpow2 85.8%

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

      4. unpow2 85.8%

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

      5. swap-sqr 93.8%

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

      6. unpow2 93.8%

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

      7. *-commutative 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in r around 0 23.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   9. Step-by-step derivation

      1. sub-neg 23.4%

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

      2. associate-*r/ 23.4%

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

      3. metadata-eval 23.4%

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

      4. metadata-eval 23.4%

         \[\leadsto \frac{2}{{r}^{2}} + \color{blue}{-1.5} \]

   10. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{2}{{r}^{2}} + -1.5} \]

   11. Taylor expanded in r around inf 23.2%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.205:\\ \;\;\;\;\frac{\frac{-2}{r}}{-r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{\frac{2}{r_m}}{r_m} + -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (+ (/ (/ 2.0 r_m) r_m) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + -1.5;
}

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((2.0 / r_m) / r_m) + -1.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(2.0 / r_m) / r_m) + -1.5)
end

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

3. Simplified 94.0%

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

5. Taylor expanded in v around inf 78.4%

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

   1. *-commutative 78.4%

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

   2. *-commutative 78.4%

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

   3. unpow2 78.4%

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

   4. unpow2 78.4%

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

   5. swap-sqr 95.1%

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

   6. unpow2 95.1%

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

   7. *-commutative 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in r around 0 56.0%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
9. Step-by-step derivation

   1. sub-neg 56.0%

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

   2. associate-*r/ 56.0%

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

   3. metadata-eval 56.0%

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

   4. metadata-eval 56.0%

      \[\leadsto \frac{2}{{r}^{2}} + \color{blue}{-1.5} \]

10. Simplified 56.0%

    \[\leadsto \color{blue}{\frac{2}{{r}^{2}} + -1.5} \]
11. Step-by-step derivation

    1. unpow2 56.0%

       \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -1.5 \]

    2. associate-/l/ 56.0%

       \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + -1.5 \]

    3. expm1-log1p-u 54.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{r}}{r}\right)\right)} + -1.5 \]

    4. expm1-udef 54.0%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{2}{r}}{r}\right)} - 1\right)} + -1.5 \]

    5. associate-/l/ 54.0%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{r \cdot r}}\right)} - 1\right) + -1.5 \]

    6. unpow2 54.0%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{r}^{2}}}\right)} - 1\right) + -1.5 \]

    7. div-inv 54.0%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}}\right)} - 1\right) + -1.5 \]

    8. pow-flip 54.0%

       \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + -1.5 \]

    9. metadata-eval 54.0%

       \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + -1.5 \]

12. Applied egg-rr 54.0%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + -1.5 \]
13. Step-by-step derivation

    1. expm1-def 54.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + -1.5 \]

    2. expm1-log1p 56.1%

       \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + -1.5 \]

14. Simplified 56.1%

    \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + -1.5 \]
15. Step-by-step derivation

    1. metadata-eval 56.1%

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

    2. pow-prod-up 55.9%

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

    3. inv-pow 55.9%

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

    4. inv-pow 55.9%

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

    5. associate-*r* 55.9%

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

    6. div-inv 55.9%

       \[\leadsto \color{blue}{\frac{2}{r}} \cdot \frac{1}{r} + -1.5 \]

    7. div-inv 56.0%

       \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + -1.5 \]

16. Applied egg-rr 56.0%

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + -1.5 \]

17. Final simplification 56.0%

    \[\leadsto \frac{\frac{2}{r}}{r} + -1.5 \]
18. Add Preprocessing

Alternative 9: 13.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 -1.5)

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -1.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = -1.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return -1.5
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = -1.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := -1.5

Derivation

1. Initial program 82.3%

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

3. Simplified 94.0%

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

5. Taylor expanded in v around inf 78.4%

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

   1. *-commutative 78.4%

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

   2. *-commutative 78.4%

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

   3. unpow2 78.4%

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

   4. unpow2 78.4%

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

   5. swap-sqr 95.1%

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

   6. unpow2 95.1%

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

   7. *-commutative 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in r around 0 56.0%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
9. Step-by-step derivation

   1. sub-neg 56.0%

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

   2. associate-*r/ 56.0%

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

   3. metadata-eval 56.0%

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

   4. metadata-eval 56.0%

      \[\leadsto \frac{2}{{r}^{2}} + \color{blue}{-1.5} \]

10. Simplified 56.0%

    \[\leadsto \color{blue}{\frac{2}{{r}^{2}} + -1.5} \]

11. Taylor expanded in r around inf 10.5%

    \[\leadsto \color{blue}{-1.5} \]

12. Final simplification 10.5%

    \[\leadsto -1.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))