Rosa's TurbineBenchmark

Percentage Accurate: 85.4% → 99.8%

Time: 15.4s

Alternatives: 5

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: 85.4% 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

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

FPCore

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

C

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

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) + (((r * w) ** 2.0d0) * (((v * (-0.25d0)) - (-0.375d0)) / (v + (-1.0d0)))))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 + (Math.pow((r * w), 2.0) * (((v * -0.25) - -0.375) / (v + -1.0))));
}

Python

def code(v, w, r):
	return ((2.0 / r) / r) + (-1.5 + (math.pow((r * w), 2.0) * (((v * -0.25) - -0.375) / (v + -1.0))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 + (((r * w) ^ 2.0) * (((v * -0.25) - -0.375) / (v + -1.0))));
end

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 94.8%

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

   1. frac-2neg 94.8%

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

   2. *-commutative 94.8%

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

   3. associate-*r* 84.8%

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

   4. div-inv 84.8%

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

   5. associate-*r* 94.8%

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

   6. *-commutative 94.8%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   4. fma-udef 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   12. associate--r- 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. associate-/r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+45} \lor \neg \left(v \leq 6.9 \cdot 10^{-12}\right):\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ (/ 2.0 r) r)))
   (if (or (<= v -1e+45) (not (<= v 6.9e-12)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* t_0 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -1e+45) || !(v <= 6.9e-12)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = (2.0d0 / r) / r
    if ((v <= (-1d+45)) .or. (.not. (v <= 6.9d-12))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.375d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -1e+45) || !(v <= 6.9e-12)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / r) / r
	tmp = 0
	if (v <= -1e+45) or not (v <= 6.9e-12):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (t_0 * 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -1e+45) || !(v <= 6.9e-12))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -1e+45) || ~((v <= 6.9e-12)))
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	else
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -1e+45], N[Not[LessEqual[v, 6.9e-12]], $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[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -9.9999999999999993e44 or 6.9000000000000001e-12 < v

   1. Initial program 82.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in v around inf 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. unpow2 81.4%

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

      4. unpow2 81.4%

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

      5. swap-sqr 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. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -9.9999999999999993e44 < v < 6.9000000000000001e-12

   1. Initial program 83.5%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in v around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. *-commutative 79.1%

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

      3. unpow2 79.1%

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

      4. unpow2 79.1%

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

      5. swap-sqr 99.4%

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

      6. unpow2 99.4%

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

      7. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. unpow2 87.2%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+45} \lor \neg \left(v \leq 6.9 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ t_2 := -1.5 - t_0 \cdot 0.25\\ \mathbf{if}\;v \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + t_2\\ \mathbf{elif}\;v \leq 6.9 \cdot 10^{-12}:\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.375\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w)))
        (t_1 (/ (/ 2.0 r) r))
        (t_2 (- -1.5 (* t_0 0.25))))
   (if (<= v -1e+44)
     (+ (/ 1.0 (* r (/ r 2.0))) t_2)
     (if (<= v 6.9e-12) (+ t_1 (- -1.5 (* t_0 0.375))) (+ t_1 t_2)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -1e+44) {
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	} else if (v <= 6.9e-12) {
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	} else {
		tmp = t_1 + t_2;
	}
	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) * (r * w)
    t_1 = (2.0d0 / r) / r
    t_2 = (-1.5d0) - (t_0 * 0.25d0)
    if (v <= (-1d+44)) then
        tmp = (1.0d0 / (r * (r / 2.0d0))) + t_2
    else if (v <= 6.9d-12) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.375d0))
    else
        tmp = t_1 + t_2
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -1e+44) {
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	} else if (v <= 6.9e-12) {
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	} else {
		tmp = t_1 + t_2;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / r) / r
	t_2 = -1.5 - (t_0 * 0.25)
	tmp = 0
	if v <= -1e+44:
		tmp = (1.0 / (r * (r / 2.0))) + t_2
	elif v <= 6.9e-12:
		tmp = t_1 + (-1.5 - (t_0 * 0.375))
	else:
		tmp = t_1 + t_2
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	t_2 = Float64(-1.5 - Float64(t_0 * 0.25))
	tmp = 0.0
	if (v <= -1e+44)
		tmp = Float64(Float64(1.0 / Float64(r * Float64(r / 2.0))) + t_2);
	elseif (v <= 6.9e-12)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.375)));
	else
		tmp = Float64(t_1 + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	t_2 = -1.5 - (t_0 * 0.25);
	tmp = 0.0;
	if (v <= -1e+44)
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	elseif (v <= 6.9e-12)
		tmp = t_1 + (-1.5 - (t_0 * 0.375));
	else
		tmp = t_1 + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$2 = N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1e+44], N[(N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[v, 6.9e-12], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.0000000000000001e44

   1. Initial program 82.9%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in v around inf 81.1%

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

      1. *-commutative 81.1%

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

      2. *-commutative 81.1%

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

      3. unpow2 81.1%

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

      4. unpow2 81.1%

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

      5. swap-sqr 99.9%

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

      6. unpow2 99.9%

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

      7. *-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. clear-num 99.8%

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

       2. inv-pow 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

       2. associate-/r/ 99.9%

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

   13. Simplified 99.9%

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

   if -1.0000000000000001e44 < v < 6.9000000000000001e-12

   1. Initial program 83.5%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in v around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. *-commutative 79.1%

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

      3. unpow2 79.1%

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

      4. unpow2 79.1%

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

      5. swap-sqr 99.4%

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

      6. unpow2 99.4%

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

      7. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. unpow2 87.2%

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

   9. Applied egg-rr 99.4%

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

   if 6.9000000000000001e-12 < v

   1. Initial program 82.1%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in v around inf 81.7%

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

      1. *-commutative 81.7%

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

      2. *-commutative 81.7%

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

      3. unpow2 81.7%

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

      4. unpow2 81.7%

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

      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. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 6.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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) + (((v * (-0.25d0)) - (-0.375d0)) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

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

Python

def code(v, w, r):
	return ((2.0 / r) / r) + (-1.5 + (((v * -0.25) - -0.375) / ((v + -1.0) / ((r * w) * (r * w)))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 + (((v * -0.25) - -0.375) / ((v + -1.0) / ((r * w) * (r * w)))));
end

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 94.8%

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

   1. frac-2neg 94.8%

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

   2. *-commutative 94.8%

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

   3. associate-*r* 84.8%

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

   4. div-inv 84.8%

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

   5. associate-*r* 94.8%

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

   6. *-commutative 94.8%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   4. fma-udef 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   12. associate--r- 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 93.0%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 93.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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) - (((r * w) * (r * w)) * 0.25d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
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 93.0%

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

   6. unpow2 93.0%

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

   7. *-commutative 93.0%

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

7. Simplified 93.0%

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

   1. unpow2 93.0%

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

9. Applied egg-rr 93.0%

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

10. Final simplification 93.0%

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

Reproduce

herbie shell --seed 2024017 
(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))