Rosa's TurbineBenchmark

Percentage Accurate: 85.6% → 99.8%

Time: 11.8s

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, 0.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

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

   1. *-commutative 96.3%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 86.5%

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

   5. un-div-inv 86.5%

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

   6. fma-udef 86.5%

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

   7. metadata-eval 86.5%

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

   8. associate-*l* 86.5%

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

   9. *-commutative 86.5%

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

   10. metadata-eval 86.5%

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

   11. distribute-rgt-in 86.5%

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

   12. +-commutative 86.5%

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

   13. clear-num 86.5%

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

5. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

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

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

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 / r) / r) + Float64(-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 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 + 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]), $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%

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

   1. *-commutative 96.3%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 86.5%

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

   5. un-div-inv 86.5%

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

   6. fma-udef 86.5%

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

   7. metadata-eval 86.5%

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

   8. associate-*l* 86.5%

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

   9. *-commutative 86.5%

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

   10. metadata-eval 86.5%

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

   11. distribute-rgt-in 86.5%

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

   12. +-commutative 86.5%

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

   13. clear-num 86.5%

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

5. Applied egg-rr 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. div-inv 99.7%

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

   4. distribute-lft-neg-in 99.7%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. neg-sub0 99.8%

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

   5. associate--r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. neg-sub0 99.8%

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

   8. fma-udef 99.8%

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

   9. *-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. associate--r+ 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-commutative 99.8%

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

9. Simplified 99.8%

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

    1. metadata-eval 99.8%

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

    2. pow-prod-up 99.8%

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

    3. unpow-1 99.8%

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

    4. unpow-1 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot w}\\ \mathbf{if}\;v \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{-1}{t_0 \cdot t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r w))))
   (if (<= v 2.3e+54)
     (+ -1.5 (+ (/ 2.0 (* r r)) (* (* (* r w) (* r w)) -0.375)))
     (+ (/ (/ 2.0 r) r) (+ -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 tmp;
	if (v <= 2.3e+54) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = ((2.0 / r) / r) + (-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) :: tmp
    t_0 = 2.0d0 / (r * w)
    if (v <= 2.3d+54) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + (((r * w) * (r * w)) * (-0.375d0)))
    else
        tmp = ((2.0d0 / r) / r) + ((-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 tmp;
	if (v <= 2.3e+54) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 + (-1.0 / (t_0 * t_0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * w)
	tmp = 0
	if v <= 2.3e+54:
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 + (-1.0 / (t_0 * t_0)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * w))
	tmp = 0.0
	if (v <= 2.3e+54)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-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);
	tmp = 0.0;
	if (v <= 2.3e+54)
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	else
		tmp = ((2.0 / r) / r) + (-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]}, If[LessEqual[v, 2.3e+54], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 2.29999999999999994e54

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in v around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. unpow2 82.7%

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

      3. unpow2 82.7%

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

      4. swap-sqr 99.0%

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

      5. unpow2 99.0%

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

   6. Simplified 99.0%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.0%

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

   if 2.29999999999999994e54 < v

   1. Initial program 75.8%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. clear-num 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-*r* 86.8%

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

      5. un-div-inv 86.8%

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

      6. fma-udef 86.8%

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

      7. metadata-eval 86.8%

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

      8. associate-*l* 86.8%

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

      9. *-commutative 86.8%

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

      10. metadata-eval 86.8%

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

      11. distribute-rgt-in 86.8%

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

      12. +-commutative 86.8%

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

      13. clear-num 86.9%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 78.5%

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

      1. unpow2 78.5%

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

      2. unpow2 78.5%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   8. Simplified 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. sqrt-div 99.8%

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

      3. metadata-eval 99.8%

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

      4. unpow2 99.8%

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

      5. sqrt-prod 63.1%

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

      6. add-sqr-sqrt 81.6%

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

      7. sqrt-div 81.6%

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

      8. metadata-eval 81.6%

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

      9. unpow2 81.6%

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

      10. sqrt-prod 63.1%

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

      11. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{-1}{\frac{2}{r \cdot w} \cdot \frac{2}{r \cdot w}}\right)\\ \end{array} \]

Alternative 5: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 6.2 \cdot 10^{+84}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r} \cdot \frac{1}{r} + -0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 6.2e+84)
   (+ -1.5 (+ (/ 2.0 (* r r)) (* (* (* r w) (* r w)) -0.375)))
   (+ -1.5 (+ (* (/ 2.0 r) (/ 1.0 r)) (* -0.25 (* r (* r (* w w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 6.2e+84) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = -1.5 + (((2.0 / r) * (1.0 / r)) + (-0.25 * (r * (r * (w * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (v <= 6.2d+84) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + (((r * w) * (r * w)) * (-0.375d0)))
    else
        tmp = (-1.5d0) + (((2.0d0 / r) * (1.0d0 / r)) + ((-0.25d0) * (r * (r * (w * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (v <= 6.2e+84) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = -1.5 + (((2.0 / r) * (1.0 / r)) + (-0.25 * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 6.2e+84:
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375))
	else:
		tmp = -1.5 + (((2.0 / r) * (1.0 / r)) + (-0.25 * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 6.2e+84)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)));
	else
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) * Float64(1.0 / r)) + Float64(-0.25 * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 6.2e+84)
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	else
		tmp = -1.5 + (((2.0 / r) * (1.0 / r)) + (-0.25 * (r * (r * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 6.2e+84], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 6.20000000000000006e84

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in v around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. unpow2 81.6%

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

      3. unpow2 81.6%

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

      4. swap-sqr 98.6%

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

      5. unpow2 98.6%

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

   6. Simplified 98.6%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 98.6%

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

   if 6.20000000000000006e84 < v

   1. Initial program 77.2%

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

   3. Simplified 89.2%

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

      1. associate-/r* 89.2%

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

      2. div-inv 89.1%

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

   5. Applied egg-rr 89.1%

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

   6. Taylor expanded in v around inf 89.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 6.2 \cdot 10^{+84}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r} \cdot \frac{1}{r} + -0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot 0.5\\ \mathbf{if}\;v \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) 0.5)))
   (if (<= v 3.2e+54)
     (+ -1.5 (+ (/ 2.0 (* r r)) (* (* (* r w) (* r w)) -0.375)))
     (+ (/ (/ 2.0 r) r) (- -1.5 (* t_0 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * 0.5;
	double tmp;
	if (v <= 3.2e+54) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (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) :: tmp
    t_0 = (r * w) * 0.5d0
    if (v <= 3.2d+54) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + (((r * w) * (r * w)) * (-0.375d0)))
    else
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * 0.5;
	double tmp;
	if (v <= 3.2e+54) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * 0.5
	tmp = 0
	if v <= 3.2e+54:
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * 0.5)
	tmp = 0.0
	if (v <= 3.2e+54)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * 0.5;
	tmp = 0.0;
	if (v <= 3.2e+54)
		tmp = -1.5 + ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.375));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[v, 3.2e+54], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 3.2e54

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in v around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. unpow2 82.7%

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

      3. unpow2 82.7%

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

      4. swap-sqr 99.0%

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

      5. unpow2 99.0%

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

   6. Simplified 99.0%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.0%

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

   if 3.2e54 < v

   1. Initial program 75.8%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. clear-num 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-*r* 86.8%

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

      5. un-div-inv 86.8%

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

      6. fma-udef 86.8%

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

      7. metadata-eval 86.8%

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

      8. associate-*l* 86.8%

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

      9. *-commutative 86.8%

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

      10. metadata-eval 86.8%

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

      11. distribute-rgt-in 86.8%

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

      12. +-commutative 86.8%

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

      13. clear-num 86.9%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 78.5%

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

      1. unpow2 78.5%

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

      2. unpow2 78.5%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   8. Simplified 99.7%

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

      1. associate-/r/ 99.7%

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

      2. add-sqr-sqrt 99.7%

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

      3. sqrt-prod 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{{\left(r \cdot w\right)}^{2}}\right) \cdot \sqrt{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\color{blue}{0.5} \cdot \sqrt{{\left(r \cdot w\right)}^{2}}\right) \cdot \sqrt{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

      6. unpow2 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \sqrt{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \cdot \sqrt{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

      7. sqrt-prod 63.1%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(\sqrt{r \cdot w} \cdot \sqrt{r \cdot w}\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

      8. add-sqr-sqrt 81.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

      9. sqrt-prod 81.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{{\left(r \cdot w\right)}^{2}}\right)}\right) \]

      10. metadata-eval 81.5%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{{\left(r \cdot w\right)}^{2}}\right)\right) \]

      11. metadata-eval 81.5%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(\color{blue}{0.5} \cdot \sqrt{{\left(r \cdot w\right)}^{2}}\right)\right) \]

      12. unpow2 81.5%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(0.5 \cdot \sqrt{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right)\right) \]

      13. sqrt-prod 63.1%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(\sqrt{r \cdot w} \cdot \sqrt{r \cdot w}\right)}\right)\right) \]

      14. add-sqr-sqrt 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right)\right) \]

   10. Applied egg-rr 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\left(0.5 \cdot \left(r \cdot w\right)\right) \cdot \left(0.5 \cdot \left(r \cdot w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot 0.5\right) \cdot \left(\left(r \cdot w\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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)) + (((r * w) * (r * w)) * (-0.375d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $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 99.8%

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

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%

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

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))