Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.8%

Time: 13.1s

Alternatives: 7

Speedup: 0.7×

• 52.4% 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: 84.7% 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

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (* (* r w) (* (* r w) (/ (fma v -0.25 0.375) (+ v -1.0)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 86.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 86.7%

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

   1. sub-neg 86.7%

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

   2. associate-*r/ 86.7%

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

   3. metadata-eval 86.7%

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

   4. metadata-eval 86.7%

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

7. Simplified 86.7%

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

   1. *-commutative 86.7%

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

   2. distribute-lft-in 86.7%

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

   3. distribute-lft-in 73.8%

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

9. Applied egg-rr 84.6%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{{\left(r \cdot w\right)}^{2}}{1 - v} \cdot \frac{v}{v \cdot 2.6666666666666665} + \frac{{\left(r \cdot w\right)}^{2}}{1 - v} \cdot \left(v \cdot -0.25\right)\right)}\right) \]

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(r \cdot w\right)}^{2}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.15 \lor \neg \left(v \leq 0.5\right):\\ \;\;\;\;t\_1 + \left(-1.5 - t\_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (pow (* r w) 2.0)) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -1.15) (not (<= v 0.5)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = pow((r * w), 2.0);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.15) || !(v <= 0.5)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (r * w) ** 2.0d0
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-1.15d0)) .or. (.not. (v <= 0.5d0))) 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

public static double code(double v, double w, double r) {
	double t_0 = Math.pow((r * w), 2.0);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.15) || !(v <= 0.5)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = math.pow((r * w), 2.0)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.15) or not (v <= 0.5):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * w) ^ 2.0
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.15) || !(v <= 0.5))
		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

function tmp_2 = code(v, w, r)
	t_0 = (r * w) ^ 2.0;
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.15) || ~((v <= 0.5)))
		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

code[v_, w_, r_] := Block[{t$95$0 = N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.15], N[Not[LessEqual[v, 0.5]], $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 < -1.1499999999999999 or 0.5 < v

   1. Initial program 82.2%

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

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 82.6%

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

      1. *-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. unpow2 82.6%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

   7. Simplified 99.4%

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

   if -1.1499999999999999 < v < 0.5

   1. Initial program 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 80.2%

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

      1. *-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. unpow2 80.2%

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

      4. swap-sqr 98.9%

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

      5. unpow2 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.15 \lor \neg \left(v \leq 0.5\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - {\left(r \cdot w\right)}^{2} \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+250}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - {\left(r \cdot w\right)}^{2} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+250)
     (+
      t_0
      (+ -1.5 (* (+ 0.375 (* v -0.25)) (* r (* (* w w) (/ r (+ v -1.0)))))))
     (+ t_0 (- -1.5 (* (pow (* r w) 2.0) 0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+250) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 - (pow((r * w), 2.0) * 0.25));
	}
	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 * w) <= 2d+250) then
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) ** 2.0d0) * 0.25d0))
    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 * w) <= 2e+250) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 - (Math.pow((r * w), 2.0) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+250:
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = t_0 + (-1.5 - (math.pow((r * w), 2.0) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+250)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64((Float64(r * w) ^ 2.0) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+250)
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = t_0 + (-1.5 - (((r * w) ^ 2.0) * 0.25));
	end
	tmp_2 = 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], 2e+250], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.9999999999999998e250

   1. Initial program 94.3%

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

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 97.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. distribute-rgt-in 97.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(v \cdot -2\right) \cdot 0.125 + 3 \cdot 0.125\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. associate-*l* 97.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. metadata-eval 97.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      5. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 1.9999999999999998e250 < (*.f64 w w)

   1. Initial program 62.4%

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

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. swap-sqr 97.3%

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

      5. unpow2 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - {\left(r \cdot w\right)}^{2} \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t\_1 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot t\_0\right)\\ \mathbf{if}\;v \leq -1 \cdot 10^{+242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq -3.15 \cdot 10^{+178}:\\ \;\;\;\;t\_1 + -1.5\\ \mathbf{elif}\;v \leq -13000000000000 \lor \neg \left(v \leq 4000\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* (* w w) (/ r (+ v -1.0)))))
        (t_1 (/ 2.0 (* r r)))
        (t_2 (+ t_1 (+ -1.5 (* (* v -0.25) t_0)))))
   (if (<= v -1e+242)
     t_2
     (if (<= v -3.15e+178)
       (+ t_1 -1.5)
       (if (or (<= v -13000000000000.0) (not (<= v 4000.0)))
         t_2
         (+ t_1 (+ -1.5 (* 0.375 t_0))))))))

C

double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double t_2 = t_1 + (-1.5 + ((v * -0.25) * t_0));
	double tmp;
	if (v <= -1e+242) {
		tmp = t_2;
	} else if (v <= -3.15e+178) {
		tmp = t_1 + -1.5;
	} else if ((v <= -13000000000000.0) || !(v <= 4000.0)) {
		tmp = t_2;
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = r * ((w * w) * (r / (v + (-1.0d0))))
    t_1 = 2.0d0 / (r * r)
    t_2 = t_1 + ((-1.5d0) + ((v * (-0.25d0)) * t_0))
    if (v <= (-1d+242)) then
        tmp = t_2
    else if (v <= (-3.15d+178)) then
        tmp = t_1 + (-1.5d0)
    else if ((v <= (-13000000000000.0d0)) .or. (.not. (v <= 4000.0d0))) then
        tmp = t_2
    else
        tmp = t_1 + ((-1.5d0) + (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double t_2 = t_1 + (-1.5 + ((v * -0.25) * t_0));
	double tmp;
	if (v <= -1e+242) {
		tmp = t_2;
	} else if (v <= -3.15e+178) {
		tmp = t_1 + -1.5;
	} else if ((v <= -13000000000000.0) || !(v <= 4000.0)) {
		tmp = t_2;
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * ((w * w) * (r / (v + -1.0)))
	t_1 = 2.0 / (r * r)
	t_2 = t_1 + (-1.5 + ((v * -0.25) * t_0))
	tmp = 0
	if v <= -1e+242:
		tmp = t_2
	elif v <= -3.15e+178:
		tmp = t_1 + -1.5
	elif (v <= -13000000000000.0) or not (v <= 4000.0):
		tmp = t_2
	else:
		tmp = t_1 + (-1.5 + (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0))))
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(t_1 + Float64(-1.5 + Float64(Float64(v * -0.25) * t_0)))
	tmp = 0.0
	if (v <= -1e+242)
		tmp = t_2;
	elseif (v <= -3.15e+178)
		tmp = Float64(t_1 + -1.5);
	elseif ((v <= -13000000000000.0) || !(v <= 4000.0))
		tmp = t_2;
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * ((w * w) * (r / (v + -1.0)));
	t_1 = 2.0 / (r * r);
	t_2 = t_1 + (-1.5 + ((v * -0.25) * t_0));
	tmp = 0.0;
	if (v <= -1e+242)
		tmp = t_2;
	elseif (v <= -3.15e+178)
		tmp = t_1 + -1.5;
	elseif ((v <= -13000000000000.0) || ~((v <= 4000.0)))
		tmp = t_2;
	else
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1e+242], t$95$2, If[LessEqual[v, -3.15e+178], N[(t$95$1 + -1.5), $MachinePrecision], If[Or[LessEqual[v, -13000000000000.0], N[Not[LessEqual[v, 4000.0]], $MachinePrecision]], t$95$2, N[(t$95$1 + N[(-1.5 + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000005e242 or -3.15000000000000015e178 < v < -1.3e13 or 4e3 < v

   1. Initial program 85.1%

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

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 91.0%

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

      1. *-commutative 91.0%

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

   7. Simplified 91.0%

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

   if -1.00000000000000005e242 < v < -3.15000000000000015e178

   1. Initial program 68.7%

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

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 68.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right) \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}}\right) \]

      2. pow3 68.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right)}^{3}}\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(v \cdot -0.25 + 0.375\right) \cdot \frac{{\left(r \cdot w\right)}^{2}}{1 - v}}\right)}^{3}}\right) \]

   7. Taylor expanded in r around 0 94.5%

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

   if -1.3e13 < v < 4e3

   1. Initial program 85.3%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 84.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+242}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{elif}\;v \leq -3.15 \cdot 10^{+178}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;v \leq -13000000000000 \lor \neg \left(v \leq 4000\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.6e-84)
     (+ t_0 -1.5)
     (+
      t_0
      (+ -1.5 (* (+ 0.375 (* v -0.25)) (* r (* (* w w) (/ r (+ v -1.0))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.6e-84) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	}
	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 (r <= 1.6d-84) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    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 (r <= 1.6e-84) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.6e-84:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.6e-84)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.6e-84)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (v + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.6e-84], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.6e-84

   1. Initial program 78.5%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 81.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right) \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}}\right) \]

      2. pow3 81.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right)}^{3}}\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(v \cdot -0.25 + 0.375\right) \cdot \frac{{\left(r \cdot w\right)}^{2}}{1 - v}}\right)}^{3}}\right) \]

   7. Taylor expanded in r around 0 72.4%

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

   if 1.6e-84 < r

   1. Initial program 96.1%

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

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. distribute-rgt-in 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(v \cdot -2\right) \cdot 0.125 + 3 \cdot 0.125\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. associate-*l* 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. metadata-eval 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      5. metadata-eval 97.2%

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

   6. Applied egg-rr 97.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.1e-83)
     (+ t_0 -1.5)
     (+ t_0 (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.1e-83) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	}
	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 (r <= 1.1d-83) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    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 (r <= 1.1e-83) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.1e-83:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.1e-83)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.1e-83)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.1e-83], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.10000000000000004e-83

   1. Initial program 78.5%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 81.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right) \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}}\right) \]

      2. pow3 81.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right)}^{3}}\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(v \cdot -0.25 + 0.375\right) \cdot \frac{{\left(r \cdot w\right)}^{2}}{1 - v}}\right)}^{3}}\right) \]

   7. Taylor expanded in r around 0 72.4%

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

   if 1.10000000000000004e-83 < r

   1. Initial program 96.1%

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

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 77.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 86.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right) \cdot \sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}}\right) \]

   2. pow3 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}\right)}^{3}}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\left(v \cdot -0.25 + 0.375\right) \cdot \frac{{\left(r \cdot w\right)}^{2}}{1 - v}}\right)}^{3}}\right) \]

7. Taylor expanded in r around 0 60.4%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))