Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 97.1%

Time: 5.3s

Alternatives: 14

Speedup: 1.3×

• 53.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    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: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;{r}^{-2} \cdot 2 - \mathsf{fma}\left(0.25, {\left(w \cdot r\right)}^{2}, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v -1.1e+55)
   (- (* (pow r -2.0) 2.0) (fma 0.25 (pow (* w r) 2.0) 1.5))
   (-
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* r w) (* r w))) (- 1.0 v)))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= -1.1e+55) {
		tmp = (pow(r, -2.0) * 2.0) - fma(0.25, pow((w * r), 2.0), 1.5);
	} else {
		tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= -1.1e+55)
		tmp = Float64(Float64((r ^ -2.0) * 2.0) - fma(0.25, (Float64(w * r) ^ 2.0), 1.5));
	else
		tmp = 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(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, -1.1e+55], N[(N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision] - N[(0.25 * N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], 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[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.10000000000000005e55

   1. Initial program 74.0%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \left(\color{blue}{\frac{3}{2}} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \left(\color{blue}{\frac{3}{2}} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto {r}^{-2} \cdot 2 - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto {r}^{-2} \cdot 2 - \left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\frac{3}{2}}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{{r}^{2} \cdot {w}^{2}}, \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\frac{1}{4}, {w}^{2} \cdot \color{blue}{{r}^{2}}, \frac{3}{2}\right) \]

      10. pow-prod-down N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\frac{1}{4}, {\left(w \cdot r\right)}^{\color{blue}{2}}, \frac{3}{2}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\frac{1}{4}, {\left(w \cdot r\right)}^{\color{blue}{2}}, \frac{3}{2}\right) \]

      12. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.10000000000000005e55 < v

   1. Initial program 89.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}\\ \mathbf{elif}\;t\_0 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_0 -1e+290)
     (/ (/ (fma (* r r) -1.5 2.0) r) r)
     (if (<= t_0 -1.5) -1.5 (/ (fma -1.5 (* r r) 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_0 <= -1e+290) {
		tmp = (fma((r * r), -1.5, 2.0) / r) / r;
	} else if (t_0 <= -1.5) {
		tmp = -1.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = 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)
	tmp = 0.0
	if (t_0 <= -1e+290)
		tmp = Float64(Float64(fma(Float64(r * r), -1.5, 2.0) / r) / r);
	elseif (t_0 <= -1.5)
		tmp = -1.5;
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+290], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5 + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[t$95$0, -1.5], -1.5, N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.00000000000000006e290

   1. Initial program 85.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 5.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 53.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]

   if -1.00000000000000006e290 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.3%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 39.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.7%

      \[\leadsto -1.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.9%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v -2e+147)
     (- (- t_0 (* 0.25 (pow (* w r) 2.0))) 4.5)
     (-
      (- t_0 (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* r w) (* r w))) (- 1.0 v)))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -2e+147) {
		tmp = (t_0 - (0.25 * pow((w * r), 2.0))) - 4.5;
	} else {
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= (-2d+147)) then
        tmp = (t_0 - (0.25d0 * ((w * r) ** 2.0d0))) - 4.5d0
    else
        tmp = (t_0 - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * ((r * w) * (r * w))) / (1.0d0 - v))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -2e+147) {
		tmp = (t_0 - (0.25 * Math.pow((w * r), 2.0))) - 4.5;
	} else {
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= -2e+147:
		tmp = (t_0 - (0.25 * math.pow((w * r), 2.0))) - 4.5
	else:
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= -2e+147)
		tmp = Float64(Float64(t_0 - Float64(0.25 * (Float64(w * r) ^ 2.0))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= -2e+147)
		tmp = (t_0 - (0.25 * ((w * r) ^ 2.0))) - 4.5;
	else
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e+147], N[(N[(t$95$0 - N[(0.25 * N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e147

   1. Initial program 63.8%

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

   3. Taylor expanded in v around inf

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      3. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot {\left(w \cdot r\right)}^{\color{blue}{2}}\right) - \frac{9}{2} \]

      4. lower-pow.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot {\left(w \cdot r\right)}^{\color{blue}{2}}\right) - \frac{9}{2} \]

      5. lower-*.f64 99.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right) - 4.5 \]

   5. Applied rewrites 99.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]

   if -2e147 < v

   1. Initial program 89.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{t\_0 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{t\_0 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* 0.125 (- 3.0 (* 2.0 v)))))
   (if (<=
        (-
         (- (+ 3.0 (/ 2.0 (* r r))) (/ (* t_0 (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5)
     (- (- 3.0 (/ (* t_0 (* (* r w) (* r w))) (- 1.0 v))) 4.5)
     (/ (fma -1.5 (* r r) 2.0) (* r r)))))

C

double code(double v, double w, double r) {
	double t_0 = 0.125 * (3.0 - (2.0 * v));
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - ((t_0 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - ((t_0 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(t_0 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_0 * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(t$95$0 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

      1. associate-/r* 90.2

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

   7. Applied rewrites 90.2%

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

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.9%

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

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1.5)
   (- (- 3.0 (/ (* (* (fma -0.25 v 0.375) (* (* w r) w)) r) (- 1.0 v))) 4.5)
   (/ (fma -1.5 (* r r) 2.0) (* r r))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * ((w * r) * w)) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (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) <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(Float64(w * r) * w)) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[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], -1.5], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   7. Step-by-step derivation

      1. associate-/r* 68.3

         \[\leadsto \left(3 - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Applied rewrites 68.3%

      \[\leadsto \left(\color{blue}{3} - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\left(\frac{3}{8} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      6. associate-*r* N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\left(\frac{3}{8} \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 71.0

          \[\leadsto \left(3 - \frac{\color{blue}{\left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)} \cdot r}{1 - v}\right) - 4.5 \]

      11. lift-*.f64 N/A

          \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      12. *-commutative N/A

          \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      13. lower-*.f64 71.0

          \[\leadsto \left(3 - \frac{\left(0.375 \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot r}{1 - v}\right) - 4.5 \]

   10. Applied rewrites 71.0%

       \[\leadsto \left(3 - \frac{\color{blue}{\left(0.375 \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}}{1 - v}\right) - 4.5 \]

   11. Taylor expanded in v around 0

       \[\leadsto \left(3 - \frac{\left(\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(3 - \frac{\left(\left(\frac{-1}{4} \cdot v + \color{blue}{\frac{3}{8}}\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

       2. lower-fma.f64 89.7

          \[\leadsto \left(3 - \frac{\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5 \]

   13. Applied rewrites 89.7%

       \[\leadsto \left(3 - \frac{\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.9%

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

Alternative 6: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_0}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{0.375 \cdot t\_0}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r)))
   (if (<=
        (-
         (-
          (+ 3.0 (/ 2.0 (* r r)))
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_0) (- 1.0 v)))
         4.5)
        -1.5)
     (- (- 3.0 (/ (* 0.375 t_0) 1.0)) 4.5)
     (/ (fma -1.5 (* r r) 2.0) (* r r)))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - ((0.375 * t_0) / 1.0)) - 4.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_0) / Float64(1.0 - v))) - 4.5) <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * t_0) / 1.0)) - 4.5);
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[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] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(0.375 * t$95$0), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   7. Step-by-step derivation

      1. associate-/r* 68.3

         \[\leadsto \left(3 - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Applied rewrites 68.3%

      \[\leadsto \left(\color{blue}{3} - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   9. Taylor expanded in v around 0

      \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{1}}\right) - \frac{9}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 84.1%

       \[\leadsto \left(3 - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{1}}\right) - 4.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.9%

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

Alternative 7: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 0.42:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot r\right) \cdot \frac{w \cdot r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= r 0.42)
     (- (- t_0 (/ (* 0.375 (* w (* (* w r) r))) 1.0)) 4.5)
     (-
      (- t_0 (* (* (* (* (fma v -2.0 3.0) 0.125) w) r) (/ (* w r) (- 1.0 v))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 0.42) {
		tmp = (t_0 - ((0.375 * (w * ((w * r) * r))) / 1.0)) - 4.5;
	} else {
		tmp = (t_0 - ((((fma(v, -2.0, 3.0) * 0.125) * w) * r) * ((w * r) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 0.42)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(w * Float64(Float64(w * r) * r))) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * r) * Float64(Float64(w * r) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 0.42], N[(N[(t$95$0 - N[(N[(0.375 * N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 0.419999999999999984

   1. Initial program 84.5%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 79.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lower-*.f64 87.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   7. Applied rewrites 87.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.5%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - 4.5 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}}{1}\right) - \frac{9}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1}\right) - \frac{9}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1}\right) - \frac{9}{2} \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}\right)}{1}\right) - \frac{9}{2} \]

       7. lift-*.f64 95.9

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(w \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot r\right)\right)}{1}\right) - 4.5 \]

   12. Applied rewrites 95.9%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - 4.5 \]

   if 0.419999999999999984 < r

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \color{blue}{\left(3 - 2 \cdot v\right)}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{2 \cdot v}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right)} \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(3 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot v\right) \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(3 + -2 \cdot v\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      15. +-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(-2 \cdot v + 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\mathsf{fma}\left(-2, v, 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      17. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      20. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{\color{blue}{1 - v}}\right) - \frac{9}{2} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right)} \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(-2 \cdot v + 3\right)} \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{w \cdot r}{1 - v}}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{w \cdot r}{1 - v}}\right) - \frac{9}{2} \]

   8. Applied rewrites 95.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot r\right) \cdot \frac{w \cdot r}{1 - v}}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    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))) * ((r * w) * (r * w))) / (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))) * ((r * w) * (r * w))) / (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))) * ((r * w) * (r * w))) / (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(r * w) * Float64(r * w))) / 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))) * ((r * w) * (r * w))) / (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[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   5. associate-*l* N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   6. pow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

   8. pow-prod-down N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   12. lower-*.f64 95.7

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

4. Applied rewrites 95.7%

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

Alternative 9: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((((fma(-2.0, v, 3.0) * 0.125) * (w * r)) * (w * 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(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(w * r)) * Float64(w * r)) / Float64(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[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   5. associate-*l* N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   6. pow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

   8. pow-prod-down N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   12. lower-*.f64 95.7

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

4. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \color{blue}{\left(3 - 2 \cdot v\right)}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{2 \cdot v}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

   8. associate-*r* N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right)} \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   11. *-commutative N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(3 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot v\right) \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   14. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(3 + -2 \cdot v\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   15. +-commutative N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(-2 \cdot v + 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   16. lower-fma.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\mathsf{fma}\left(-2, v, 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

   20. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - 4.5 \]
7. Add Preprocessing

Alternative 10: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5900000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5900000.0)
   (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* 0.375 (* w (* (* w r) r))) 1.0)) 4.5)
   (- (- 3.0 (/ (* (* (fma -0.25 v 0.375) (* (* w r) w)) r) (- 1.0 v))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5900000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * (w * ((w * r) * r))) / 1.0)) - 4.5;
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * ((w * r) * w)) * r) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5900000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.375 * Float64(w * Float64(Float64(w * r) * r))) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(Float64(w * r) * w)) * r) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5900000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.9e6

   1. Initial program 84.7%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lower-*.f64 87.7

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   7. Applied rewrites 87.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.6%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - 4.5 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}}{1}\right) - \frac{9}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1}\right) - \frac{9}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1}\right) - \frac{9}{2} \]

       4. associate-*l* N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}\right)}{1}\right) - \frac{9}{2} \]

       7. lift-*.f64 95.9

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(w \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot r\right)\right)}{1}\right) - 4.5 \]

   12. Applied rewrites 95.9%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1}\right) - 4.5 \]

   if 5.9e6 < r

   1. Initial program 92.6%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   7. Step-by-step derivation

      1. associate-/r* 75.2

         \[\leadsto \left(3 - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Applied rewrites 75.2%

      \[\leadsto \left(\color{blue}{3} - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\frac{3}{8} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\left(\frac{3}{8} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      6. associate-*r* N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(3 - \frac{\color{blue}{\left(\frac{3}{8} \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 77.0

          \[\leadsto \left(3 - \frac{\color{blue}{\left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)} \cdot r}{1 - v}\right) - 4.5 \]

      11. lift-*.f64 N/A

          \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      12. *-commutative N/A

          \[\leadsto \left(3 - \frac{\left(\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      13. lower-*.f64 77.0

          \[\leadsto \left(3 - \frac{\left(0.375 \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot r}{1 - v}\right) - 4.5 \]

   10. Applied rewrites 77.0%

       \[\leadsto \left(3 - \frac{\color{blue}{\left(0.375 \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}}{1 - v}\right) - 4.5 \]

   11. Taylor expanded in v around 0

       \[\leadsto \left(3 - \frac{\left(\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(3 - \frac{\left(\left(\frac{-1}{4} \cdot v + \color{blue}{\frac{3}{8}}\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

       2. lower-fma.f64 95.4

          \[\leadsto \left(3 - \frac{\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5 \]

   13. Applied rewrites 95.4%

       \[\leadsto \left(3 - \frac{\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot r}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* 0.375 (* (* w r) (* w r))) 1.0)) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.375 * ((w * r) * (w * r))) / 1.0)) - 4.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - ((0.375d0 * ((w * r) * (w * r))) / 1.0d0)) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.375 * ((w * r) * (w * r))) / 1.0)) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - ((0.375 * ((w * r) * (w * r))) / 1.0)) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.375 * Float64(Float64(w * r) * Float64(w * r))) / 1.0)) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * ((w * r) * (w * r))) / 1.0)) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
4. Step-by-step derivation

5. Applied rewrites 78.8%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   7. lower-*.f64 85.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

7. Applied rewrites 85.0%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

8. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - \frac{9}{2} \]
9. Step-by-step derivation

10. Applied rewrites 93.6%

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1}}\right) - 4.5 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}}{1}\right) - \frac{9}{2} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1}\right) - \frac{9}{2} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1}\right) - \frac{9}{2} \]

    4. associate-*r* N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1}\right) - \frac{9}{2} \]

    5. pow2 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1}\right) - \frac{9}{2} \]

    6. associate-*l* N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

    7. pow2 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1}\right) - \frac{9}{2} \]

    8. unpow-prod-down N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{{\left(w \cdot r\right)}^{2}}}{1}\right) - \frac{9}{2} \]

    9. pow2 N/A

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1}\right) - \frac{9}{2} \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\frac{3}{8} \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1}\right) - \frac{9}{2} \]

    12. lift-*.f64 95.1

        \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1}\right) - 4.5 \]

12. Applied rewrites 95.1%

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1}\right) - 4.5 \]
13. Add Preprocessing

Alternative 12: 54.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6200000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 6200000.0) (/ (fma -1.5 (* r r) 2.0) (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 6200000.0) {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 6200000.0)
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 6200000.0], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 6.2e6

   1. Initial program 84.7%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 69.4

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   if 6.2e6 < r

   1. Initial program 92.6%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 15.5

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.2%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 49.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00013:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.00013) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00013) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00013d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00013) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00013:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00013)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00013)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00013], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.29999999999999989e-4

   1. Initial program 84.5%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 69.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 62.1%

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

   if 1.29999999999999989e-4 < r

   1. Initial program 92.9%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 16.4

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.4%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 14.0% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(v, w, r):
	return -1.5

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   6. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   7. lift-*.f64 55.7

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

5. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

6. Taylor expanded in r around inf

   \[\leadsto \frac{-3}{2} \]
7. Step-by-step derivation

8. Applied rewrites 14.9%

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

Reproduce

herbie shell --seed 2025071 
(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))