Rosa's TurbineBenchmark

Percentage Accurate: 85.6% → 99.8%

Time: 12.3s

Alternatives: 7

Speedup: 1.7×

• 53.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.3%

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

   1. *-commutative 96.3%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 86.5%

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

   5. un-div-inv 86.5%

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

   6. fma-udef 86.5%

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

   7. metadata-eval 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{v \cdot \color{blue}{\left(-2 \cdot 0.125\right)} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   8. associate-*l* 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(v \cdot -2\right) \cdot 0.125} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   9. *-commutative 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(-2 \cdot v\right)} \cdot 0.125 + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   10. metadata-eval 86.5%

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

   11. distribute-rgt-in 86.5%

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

   12. +-commutative 86.5%

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

   13. clear-num 86.5%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -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 + (v * (-2.0d0)))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))) + (-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 + (v * -2.0))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -4.5;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 86.5%

   \[\leadsto \color{blue}{\left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right)\right) + -4.5} \]
4. Step-by-step derivation

   1. associate-*r* 96.3%

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

   2. *-commutative 96.3%

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

   3. *-un-lft-identity 96.3%

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

   4. associate-*r* 99.8%

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

   5. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{r \cdot w}\\ \left(\frac{2}{r \cdot r} + -1.5\right) + \frac{-1}{\frac{\left(t_0 \cdot t_0\right) \cdot \left(v + -1\right)}{-0.375 - v \cdot -0.25}} \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* r w))))
   (+
    (+ (/ 2.0 (* r r)) -1.5)
    (/ -1.0 (/ (* (* t_0 t_0) (+ v -1.0)) (- -0.375 (* v -0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	return ((2.0 / (r * r)) + -1.5) + (-1.0 / (((t_0 * t_0) * (v + -1.0)) / (-0.375 - (v * -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
    real(8) :: t_0
    t_0 = 1.0d0 / (r * w)
    code = ((2.0d0 / (r * r)) + (-1.5d0)) + ((-1.0d0) / (((t_0 * t_0) * (v + (-1.0d0))) / ((-0.375d0) - (v * (-0.25d0)))))
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	return ((2.0 / (r * r)) + -1.5) + (-1.0 / (((t_0 * t_0) * (v + -1.0)) / (-0.375 - (v * -0.25))));
}

Python

def code(v, w, r):
	t_0 = 1.0 / (r * w)
	return ((2.0 / (r * r)) + -1.5) + (-1.0 / (((t_0 * t_0) * (v + -1.0)) / (-0.375 - (v * -0.25))))

Julia

function code(v, w, r)
	t_0 = Float64(1.0 / Float64(r * w))
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) + Float64(-1.0 / Float64(Float64(Float64(t_0 * t_0) * Float64(v + -1.0)) / Float64(-0.375 - Float64(v * -0.25)))))
end

MATLAB

function tmp = code(v, w, r)
	t_0 = 1.0 / (r * w);
	tmp = ((2.0 / (r * r)) + -1.5) + (-1.0 / (((t_0 * t_0) * (v + -1.0)) / (-0.375 - (v * -0.25))));
end

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.3%

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

   1. *-commutative 96.3%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 86.5%

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

   5. un-div-inv 86.5%

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

   6. fma-udef 86.5%

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

   7. metadata-eval 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{v \cdot \color{blue}{\left(-2 \cdot 0.125\right)} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   8. associate-*l* 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(v \cdot -2\right) \cdot 0.125} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   9. *-commutative 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(-2 \cdot v\right)} \cdot 0.125 + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   10. metadata-eval 86.5%

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

   11. distribute-rgt-in 86.5%

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

   12. +-commutative 86.5%

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

   13. clear-num 86.5%

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

5. Applied egg-rr 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. div-inv 99.7%

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

   4. distribute-lft-neg-in 99.7%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. neg-sub0 99.8%

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

   5. associate--r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. neg-sub0 99.8%

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

   8. fma-udef 99.8%

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

   9. *-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. *-commutative 99.8%

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

   12. associate--r+ 99.8%

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

   13. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. metadata-eval 99.8%

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

    2. pow-prod-up 99.8%

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

    3. unpow-1 99.8%

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

    4. unpow-1 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + -1.5) + ((r * w) * ((((v * -0.25) - -0.375) / (1.0 / (r * w))) / (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) * ((((v * (-0.25d0)) - (-0.375d0)) / (1.0d0 / (r * w))) / (v + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.3%

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

   1. *-commutative 96.3%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 86.5%

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

   5. un-div-inv 86.5%

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

   6. fma-udef 86.5%

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

   7. metadata-eval 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{v \cdot \color{blue}{\left(-2 \cdot 0.125\right)} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   8. associate-*l* 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(v \cdot -2\right) \cdot 0.125} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   9. *-commutative 86.5%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(-2 \cdot v\right)} \cdot 0.125 + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

   10. metadata-eval 86.5%

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

   11. distribute-rgt-in 86.5%

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

   12. +-commutative 86.5%

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

   13. clear-num 86.5%

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

5. Applied egg-rr 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. div-inv 99.7%

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

   4. distribute-lft-neg-in 99.7%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. neg-sub0 99.8%

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

   5. associate--r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. neg-sub0 99.8%

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

   8. fma-udef 99.8%

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

   9. *-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. *-commutative 99.8%

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

   12. associate--r+ 99.8%

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

   13. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. metadata-eval 99.8%

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

    2. pow-prod-up 99.8%

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

    3. unpow-1 99.8%

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

    4. unpow-1 99.8%

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

11. Applied egg-rr 99.8%

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

    1. clear-num 99.8%

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

    2. *-un-lft-identity 99.8%

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

    3. associate-*l* 99.8%

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

    4. times-frac 99.8%

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

    5. clear-num 99.8%

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

    6. /-rgt-identity 99.8%

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

    7. associate-/r* 99.8%

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

13. Applied egg-rr 99.8%

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

    1. associate-/r* 98.3%

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

    2. associate-/l/ 98.3%

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

    3. *-commutative 98.3%

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

15. Simplified 98.3%

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

16. Final simplification 98.3%

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

Alternative 5: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot w}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;-1.5 + \left(t_1 + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + -1.5\right) + \frac{-1}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v 1.1e-29)
     (+ -1.5 (+ t_1 (* -0.375 (* (* r w) (* r w)))))
     (+ (+ t_1 -1.5) (/ -1.0 (* t_0 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= 1.1e-29) {
		tmp = -1.5 + (t_1 + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = (t_1 + -1.5) + (-1.0 / (t_0 * 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 = 2.0d0 / (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= 1.1d-29) then
        tmp = (-1.5d0) + (t_1 + ((-0.375d0) * ((r * w) * (r * w))))
    else
        tmp = (t_1 + (-1.5d0)) + ((-1.0d0) / (t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= 1.1e-29) {
		tmp = -1.5 + (t_1 + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = (t_1 + -1.5) + (-1.0 / (t_0 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= 1.1e-29:
		tmp = -1.5 + (t_1 + (-0.375 * ((r * w) * (r * w))))
	else:
		tmp = (t_1 + -1.5) + (-1.0 / (t_0 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 1.1e-29)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(Float64(t_1 + -1.5) + Float64(-1.0 / Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= 1.1e-29)
		tmp = -1.5 + (t_1 + (-0.375 * ((r * w) * (r * w))));
	else
		tmp = (t_1 + -1.5) + (-1.0 / (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 1.1e-29], N[(-1.5 + N[(t$95$1 + N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + -1.5), $MachinePrecision] + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < 1.09999999999999995e-29

   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 85.1%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. unpow2 80.9%

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

      3. unpow2 80.9%

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

      4. swap-sqr 98.9%

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

      5. unpow2 98.9%

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

   6. Simplified 98.9%

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

      1. unpow2 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 1.09999999999999995e-29 < v

   1. Initial program 81.3%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r* 89.2%

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

      5. un-div-inv 89.2%

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

      6. fma-udef 89.2%

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

      7. metadata-eval 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{v \cdot \color{blue}{\left(-2 \cdot 0.125\right)} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      8. associate-*l* 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(v \cdot -2\right) \cdot 0.125} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      9. *-commutative 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(-2 \cdot v\right)} \cdot 0.125 + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      10. metadata-eval 89.2%

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

      11. distribute-rgt-in 89.2%

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

      12. +-commutative 89.2%

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

      13. clear-num 89.3%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 83.2%

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

      1. unpow2 83.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{4}{\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}}} \]

      2. unpow2 83.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{4}{\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}}} \]

      3. swap-sqr 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{4}{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}} \]

      4. unpow2 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{4}{\color{blue}{{\left(r \cdot w\right)}^{2}}}} \]

   8. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\color{blue}{\sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}}} \]

      2. sqrt-div 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\color{blue}{\frac{\sqrt{4}}{\sqrt{{\left(r \cdot w\right)}^{2}}}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}} \]

      3. metadata-eval 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{\color{blue}{2}}{\sqrt{{\left(r \cdot w\right)}^{2}}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}} \]

      4. unpow2 99.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{2}{\sqrt{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}} \]

      5. sqrt-prod 61.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{2}{\color{blue}{\sqrt{r \cdot w} \cdot \sqrt{r \cdot w}}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}} \]

      6. add-sqr-sqrt 81.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{2}{\color{blue}{r \cdot w}} \cdot \sqrt{\frac{4}{{\left(r \cdot w\right)}^{2}}}} \]

      7. sqrt-div 81.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{1}{\frac{2}{r \cdot w} \cdot \color{blue}{\frac{\sqrt{4}}{\sqrt{{\left(r \cdot w\right)}^{2}}}}} \]

      8. metadata-eval 81.8%

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

      9. unpow2 81.8%

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

      10. sqrt-prod 61.3%

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

      11. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \frac{-1}{\frac{2}{r \cdot w} \cdot \frac{2}{r \cdot w}}\\ \end{array} \]

Alternative 6: 96.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v 1e-30)
     (+ -1.5 (+ t_1 (* -0.375 t_0)))
     (- (+ t_1 -1.5) (* t_0 0.25)))))

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-30) {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	} else {
		tmp = (t_1 + -1.5) - (t_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) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= 1d-30) then
        tmp = (-1.5d0) + (t_1 + ((-0.375d0) * t_0))
    else
        tmp = (t_1 + (-1.5d0)) - (t_0 * 0.25d0)
    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-30) {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	} else {
		tmp = (t_1 + -1.5) - (t_0 * 0.25);
	}
	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-30:
		tmp = -1.5 + (t_1 + (-0.375 * t_0))
	else:
		tmp = (t_1 + -1.5) - (t_0 * 0.25)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 1e-30)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * t_0)));
	else
		tmp = Float64(Float64(t_1 + -1.5) - Float64(t_0 * 0.25));
	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-30)
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	else
		tmp = (t_1 + -1.5) - (t_0 * 0.25);
	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[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 1e-30], N[(-1.5 + N[(t$95$1 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + -1.5), $MachinePrecision] - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < 1e-30

   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 85.1%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. *-commutative 80.9%

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

      2. unpow2 80.9%

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

      3. unpow2 80.9%

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

      4. swap-sqr 98.9%

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

      5. unpow2 98.9%

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

   6. Simplified 98.9%

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

      1. unpow2 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 1e-30 < v

   1. Initial program 81.3%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r* 89.2%

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

      5. un-div-inv 89.2%

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

      6. fma-udef 89.2%

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

      7. metadata-eval 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{v \cdot \color{blue}{\left(-2 \cdot 0.125\right)} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      8. associate-*l* 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(v \cdot -2\right) \cdot 0.125} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      9. *-commutative 89.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(-2 \cdot v\right)} \cdot 0.125 + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}} \]

      10. metadata-eval 89.2%

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

      11. distribute-rgt-in 89.2%

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

      12. +-commutative 89.2%

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

      13. clear-num 89.3%

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

   5. Applied egg-rr 99.8%

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

      1. frac-2neg 99.8%

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

      2. div-inv 99.8%

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

      3. div-inv 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. neg-sub0 99.8%

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

      8. fma-udef 99.8%

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

      9. *-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. *-commutative 99.8%

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

      12. associate--r+ 99.8%

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

      13. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in v around inf 83.2%

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

       1. *-commutative 83.2%

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

       2. unpow2 83.2%

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

       3. unpow2 83.2%

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

       4. swap-sqr 99.8%

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

       5. unpow2 99.8%

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

   12. Simplified 99.8%

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

       1. unpow2 86.5%

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

   14. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

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

Alternative 7: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -1.5 + ((2.0 / (r * r)) + (-0.375 * ((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 = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.375d0) * ((r * w) * (r * w))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 86.5%

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

4. Taylor expanded in v around 0 79.2%

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

   1. *-commutative 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

   4. swap-sqr 94.9%

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

   5. unpow2 94.9%

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

6. Simplified 94.9%

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

   1. unpow2 94.9%

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

8. Applied egg-rr 94.9%

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

9. Final simplification 94.9%

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

Reproduce

herbie shell --seed 2023333 
(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))