Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.3%

Time: 5.4s

Alternatives: 16

Speedup: 1.6×

• 53.2% 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: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ \mathbf{if}\;v \leq -620000000000:\\ \;\;\;\;\left(t\_0 - \frac{0.25}{{\left(w \cdot r\right)}^{-2}}\right) - 4.5\\ \mathbf{elif}\;v \leq 510000000:\\ \;\;\;\;{r}^{-2} \cdot 2 - \mathsf{fma}\left(t\_1, 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot t\_1\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))) (t_1 (* (* w r) (* w r))))
   (if (<= v -620000000000.0)
     (- (- t_0 (/ 0.25 (pow (* w r) -2.0))) 4.5)
     (if (<= v 510000000.0)
       (- (* (pow r -2.0) 2.0) (fma t_1 0.375 1.5))
       (- (- t_0 (* 0.25 t_1)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (w * r) * (w * r);
	double tmp;
	if (v <= -620000000000.0) {
		tmp = (t_0 - (0.25 / pow((w * r), -2.0))) - 4.5;
	} else if (v <= 510000000.0) {
		tmp = (pow(r, -2.0) * 2.0) - fma(t_1, 0.375, 1.5);
	} else {
		tmp = (t_0 - (0.25 * t_1)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(w * r) * Float64(w * r))
	tmp = 0.0
	if (v <= -620000000000.0)
		tmp = Float64(Float64(t_0 - Float64(0.25 / (Float64(w * r) ^ -2.0))) - 4.5);
	elseif (v <= 510000000.0)
		tmp = Float64(Float64((r ^ -2.0) * 2.0) - fma(t_1, 0.375, 1.5));
	else
		tmp = Float64(Float64(t_0 - Float64(0.25 * t_1)) - 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]}, Block[{t$95$1 = N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -620000000000.0], N[(N[(t$95$0 - N[(0.25 / N[Power[N[(w * r), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 510000000.0], N[(N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision] - N[(t$95$1 * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.2e11

   1. Initial program 78.6%

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

   3. Taylor expanded in v around 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 99.7

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

   5. Applied rewrites 99.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 99.8

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \frac{1}{{\left(w \cdot r\right)}^{-2}}\right) - 4.5 \]

   7. Applied rewrites 99.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \frac{1}{\color{blue}{{\left(w \cdot r\right)}^{-2}}}\right) - 4.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -6.2e11 < v < 5.1e8

   1. Initial program 87.8%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({w}^{2} \cdot {r}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      11. pow-prod-down N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      13. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      3. unpow2 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 5.1e8 < v

   1. Initial program 72.5%

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

   3. Taylor expanded in v around 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ t_2 := \left(t\_0 - \frac{t\_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_2 \leq 10^{+71}:\\ \;\;\;\;\left(t\_0 - \frac{t\_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;{r}^{-2} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2 (- (- t_0 (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_2 1e+71)
       (- (- t_0 (/ (* t_1 (* (* w (* w r)) r)) (- 1.0 v))) 4.5)
       (* (pow r -2.0) 2.0)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= 1e+71) {
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = pow(r, -2.0) * 2.0;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= 1e+71) {
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = Math.pow(r, -2.0) * 2.0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = 0.125 * (3.0 - (2.0 * v))
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5
	elif t_2 <= 1e+71:
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = math.pow(r, -2.0) * 2.0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_2 <= 1e+71)
		tmp = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64((r ^ -2.0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = 0.125 * (3.0 - (2.0 * v));
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	elseif (t_2 <= 1e+71)
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = (r ^ -2.0) * 2.0;
	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]}, Block[{t$95$1 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 - N[(N[(t$95$1 * 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$2, (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, 1e+71], N[(N[(t$95$0 - N[(N[(t$95$1 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.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)) < 1e71

   1. Initial program 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

   if 1e71 < (-.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 75.5%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

         \[\leadsto \frac{2 \cdot 1}{{\color{blue}{r}}^{2}} \]

      2. associate-*r/ N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{1}{{r}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot \color{blue}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot \color{blue}{2} \]

      5. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 \]

      6. metadata-eval N/A

         \[\leadsto {r}^{-2} \cdot 2 \]

      7. lower-pow.f64 100.0

         \[\leadsto {r}^{-2} \cdot 2 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ t_2 := \left(t\_0 - \frac{t\_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_2 \leq 10^{+94}:\\ \;\;\;\;\left(t\_0 - \frac{t\_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2 (- (- t_0 (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_2 1e+94)
       (- (- t_0 (/ (* t_1 (* (* w (* w r)) r)) (- 1.0 v))) 4.5)
       (/ (/ 2.0 r) r)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= 1e+94) {
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 / r) / r;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= 1e+94) {
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 / r) / r;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = 0.125 * (3.0 - (2.0 * v))
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5
	elif t_2 <= 1e+94:
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = (2.0 / r) / r
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_2 <= 1e+94)
		tmp = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(2.0 / r) / r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = 0.125 * (3.0 - (2.0 * v));
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	elseif (t_2 <= 1e+94)
		tmp = (t_0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = (2.0 / r) / r;
	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]}, Block[{t$95$1 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 - N[(N[(t$95$1 * 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$2, (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, 1e+94], N[(N[(t$95$0 - N[(N[(t$95$1 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.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)) < 1e94

   1. Initial program 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   if 1e94 < (-.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 74.6%

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

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   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 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{2}{r}}{r} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{2}{r}}{r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq 10^{+94}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq 10^{+94}:\\ \;\;\;\;\left(t\_0 - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1
         (-
          (-
           t_0
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_1 1e+94)
       (-
        (-
         t_0
         (/ (* (* (* (fma -2.0 v 3.0) 0.125) (* w (* w r))) r) (- 1.0 v)))
        4.5)
       (/ (/ 2.0 r) r)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_1 <= 1e+94) {
		tmp = (t_0 - ((((fma(-2.0, v, 3.0) * 0.125) * (w * (w * r))) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 / r) / r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - 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_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_1 <= 1e+94)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(w * Float64(w * r))) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(2.0 / r) / r);
	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]}, Block[{t$95$1 = N[(N[(t$95$0 - 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$1, (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+94], N[(N[(t$95$0 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.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)) < 1e94

   1. Initial program 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 95.8

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

   6. Applied rewrites 95.8%

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

   if 1e94 < (-.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 74.6%

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

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   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 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{2}{r}}{r} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{2}{r}}{r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq 10^{+94}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ t_2 := \left(t\_0 - \frac{t\_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_2 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{t\_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2 (- (- t_0 (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_2 -1.5)
       (- (- 3.0 (/ (* t_1 (* (* w (* w r)) r)) (- 1.0 v))) 4.5)
       (- (* (/ 1.0 (* r r)) 2.0) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = 0.125 * (3.0 - (2.0 * v))
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5
	elif t_2 <= -1.5:
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_2 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_1 * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = 0.125 * (3.0 - (2.0 * v));
	t_2 = (t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	elseif (t_2 <= -1.5)
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	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]}, Block[{t$95$1 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 - N[(N[(t$95$1 * 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$2, (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, -1.5], N[(N[(3.0 - N[(N[(t$95$1 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.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 82.0%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 94.9

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

   7. Applied rewrites 94.9%

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

   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 78.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 99.4

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 99.4

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1
         (-
          (-
           t_0
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_1 -1.5)
       (-
        (-
         3.0
         (/ (* (* (* (fma -2.0 v 3.0) 0.125) (* w (* w r))) r) (- 1.0 v)))
        4.5)
       (- (* (/ 1.0 (* r r)) 2.0) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - ((((fma(-2.0, v, 3.0) * 0.125) * (w * (w * r))) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - 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_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(w * Float64(w * r))) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	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]}, Block[{t$95$1 = N[(N[(t$95$0 - 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$1, (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.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 82.0%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 82.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 94.9

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

   7. Applied rewrites 94.9%

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

   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 78.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 99.4

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 99.4

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t\_1 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_1 - 0.25 \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right) - 4.5\\ \mathbf{elif}\;t\_2 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{0.375 \cdot t\_0}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1 (+ 3.0 (/ 2.0 (* r r))))
        (t_2
         (- (- t_1 (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_0) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (- t_1 (* 0.25 (* w (* (* r r) w)))) 4.5)
     (if (<= t_2 -1.5)
       (- (- 3.0 (/ (* 0.375 t_0) (- 1.0 v))) 4.5)
       (- (* (/ 1.0 (* r r)) 2.0) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 3.0 + (2.0 / (r * r));
	double t_2 = (t_1 - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_1 - (0.25 * (w * ((r * r) * w)))) - 4.5;
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 3.0 + (2.0 / (r * r));
	double t_2 = (t_1 - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_1 - (0.25 * (w * ((r * r) * w)))) - 4.5;
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	t_1 = 3.0 + (2.0 / (r * r))
	t_2 = (t_1 - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_1 - (0.25 * (w * ((r * r) * w)))) - 4.5
	elif t_2 <= -1.5:
		tmp = (3.0 - ((0.375 * t_0) / (1.0 - v))) - 4.5
	else:
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_2 = Float64(Float64(t_1 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_0) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_1 - Float64(0.25 * Float64(w * Float64(Float64(r * r) * w)))) - 4.5);
	elseif (t_2 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * t_0) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	t_1 = 3.0 + (2.0 / (r * r));
	t_2 = (t_1 - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_1 - (0.25 * (w * ((r * r) * w)))) - 4.5;
	elseif (t_2 <= -1.5)
		tmp = (3.0 - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	else
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - 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]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$1 - N[(0.25 * N[(w * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, -1.5], N[(N[(3.0 - N[(N[(0.375 * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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)) < -inf.0

   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 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) - 4.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 97.2

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in w around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 95.2

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

   12. Applied rewrites 95.2%

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

   if -inf.0 < (-.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 82.0%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 71.0%

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

   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 78.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 99.4

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 99.4

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ \mathbf{if}\;v \leq -620000000000 \lor \neg \left(v \leq 510000000\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;{r}^{-2} \cdot 2 - \mathsf{fma}\left(t\_0, 0.375, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w r) (* w r))))
   (if (or (<= v -620000000000.0) (not (<= v 510000000.0)))
     (- (- (+ 3.0 (/ 2.0 (* r r))) (* 0.25 t_0)) 4.5)
     (- (* (pow r -2.0) 2.0) (fma t_0 0.375 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * (w * r);
	double tmp;
	if ((v <= -620000000000.0) || !(v <= 510000000.0)) {
		tmp = ((3.0 + (2.0 / (r * r))) - (0.25 * t_0)) - 4.5;
	} else {
		tmp = (pow(r, -2.0) * 2.0) - fma(t_0, 0.375, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * Float64(w * r))
	tmp = 0.0
	if ((v <= -620000000000.0) || !(v <= 510000000.0))
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(0.25 * t_0)) - 4.5);
	else
		tmp = Float64(Float64((r ^ -2.0) * 2.0) - fma(t_0, 0.375, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -620000000000.0], N[Not[LessEqual[v, 510000000.0]], $MachinePrecision]], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision] - N[(t$95$0 * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.2e11 or 5.1e8 < v

   1. Initial program 75.6%

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

   3. Taylor expanded in v around 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -6.2e11 < v < 5.1e8

   1. Initial program 87.8%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({w}^{2} \cdot {r}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      11. pow-prod-down N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      13. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{3}{8}, \frac{3}{2}\right) \]

      3. unpow2 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{3}{8}, \frac{3}{2}\right) \]

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

      \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), 0.375, 1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -620000000000 \lor \neg \left(v \leq 510000000\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;{r}^{-2} \cdot 2 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), 0.375, 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 0.6× speedup

Math

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

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 v))) 4.5)
     (- (* (/ 1.0 (* r r)) 2.0) 1.5))))

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 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 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) :: t_0
    real(8) :: tmp
    t_0 = ((w * w) * r) * r
    if ((((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * t_0) / (1.0d0 - v))) - 4.5d0) <= (-1.5d0)) then
        tmp = (3.0d0 - ((0.375d0 * t_0) / (1.0d0 - v))) - 4.5d0
    else
        tmp = ((1.0d0 / (r * r)) * 2.0d0) - 1.5d0
    end if
    code = tmp
end function

Java

public static 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 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	tmp = 0
	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 - v))) - 4.5
	else:
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5
	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) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	tmp = 0.0;
	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 - v))) - 4.5;
	else
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	end
	tmp_2 = 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] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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 83.7%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 67.8%

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

   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 78.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 99.4

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 99.4

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

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)
      -1e+179)
   (/ (/ (fma (* r r) -1.5 2.0) r) r)
   (- (* (/ 1.0 (* r r)) 2.0) 1.5)))

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) <= -1e+179) {
		tmp = (fma((r * r), -1.5, 2.0) / r) / r;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	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) <= -1e+179)
		tmp = Float64(Float64(fma(Float64(r * r), -1.5, 2.0) / r) / r);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	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], -1e+179], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5 + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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)) < -9.9999999999999998e178

   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 3.1

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   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 50.1

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

   7. Applied rewrites 50.1%

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

   if -9.9999999999999998e178 < (-.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 78.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 85.5

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 85.5

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 85.5%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

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)
      -2e+85)
   (/ (/ (* (* r r) -1.5) r) r)
   (- (* (/ 1.0 (* r r)) 2.0) 1.5)))

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) <= -2e+85) {
		tmp = (((r * r) * -1.5) / r) / r;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 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 ((((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-2d+85)) then
        tmp = (((r * r) * (-1.5d0)) / r) / r
    else
        tmp = ((1.0d0 / (r * r)) * 2.0d0) - 1.5d0
    end if
    code = tmp
end function

Java

public static 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) <= -2e+85) {
		tmp = (((r * r) * -1.5) / r) / r;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -2e+85:
		tmp = (((r * r) * -1.5) / r) / r
	else:
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5
	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) <= -2e+85)
		tmp = Float64(Float64(Float64(Float64(r * r) * -1.5) / r) / r);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -2e+85)
		tmp = (((r * r) * -1.5) / r) / r;
	else
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	end
	tmp_2 = 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], -2e+85], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $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)) < -2e85

   1. Initial program 86.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 3.1

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   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 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in r around inf

      \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2}}{r}}{r} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 46.5

         \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]

   10. Applied rewrites 46.5%

       \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]

   if -2e85 < (-.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 78.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 88.4

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

      6. lift-*.f64 88.3

         \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

   7. Applied rewrites 88.3%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r r))) (* 0.25 (* (* w r) (* w r)))) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 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))) - (0.25d0 * ((w * r) * (w * r)))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in v around 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 94.4

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

5. Applied rewrites 94.4%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 94.4

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

7. Applied rewrites 94.4%

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

Alternative 13: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(w \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r r))) (* 0.25 (* w (* r (* w r))))) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * (w * (r * (w * r))))) - 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))) - (0.25d0 * (w * (r * (w * r))))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * (w * (r * (w * r))))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (0.25 * (w * (r * (w * r))))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(0.25 * Float64(w * Float64(r * Float64(w * r))))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (0.25 * (w * (r * (w * r))))) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in v around 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 94.4

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

5. Applied rewrites 94.4%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 94.4

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

7. Applied rewrites 94.4%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) - 4.5 \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 92.5

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

9. Applied rewrites 92.5%

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

Alternative 14: 57.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r \cdot r} \cdot 2 - 1.5 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 52.1

      \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

5. Applied rewrites 52.1%

   \[\leadsto \color{blue}{{r}^{-2} \cdot 2 - 1.5} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

      \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{{r}^{2}} \cdot 2 - \frac{3}{2} \]

   5. pow2 N/A

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - \frac{3}{2} \]

   6. lift-*.f64 52.1

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]

7. Applied rewrites 52.1%

   \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
8. Add Preprocessing

Alternative 15: 50.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.02:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.02) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.02) {
		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 <= 0.02d0) 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 <= 0.02) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.02:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.02)
		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 <= 0.02)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.02], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.0200000000000000004

   1. Initial program 78.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 55.6%

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

   if 0.0200000000000000004 < r

   1. Initial program 87.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 20.7

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

   5. Applied rewrites 20.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 24.5%

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

Alternative 16: 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 81.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 14.4%

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

Reproduce

herbie shell --seed 2025084 
(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))