Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.7%

Time: 4.6s

Alternatives: 13

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Applied rewrites 99.7%

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

Alternative 2: 95.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 35000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(v, -2, 3\right), 0.125 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot \frac{r}{1 - v}\right), 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 35000000.0)
   (- (/ 2.0 (* r r)) (fma 0.25 (* (* (* w r) r) w) 1.5))
   (-
    3.0
    (fma (fma v -2.0 3.0) (* 0.125 (* (* (* w r) w) (/ r (- 1.0 v)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 35000000.0) {
		tmp = (2.0 / (r * r)) - fma(0.25, (((w * r) * r) * w), 1.5);
	} else {
		tmp = 3.0 - fma(fma(v, -2.0, 3.0), (0.125 * (((w * r) * w) * (r / (1.0 - v)))), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 35000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(0.25, Float64(Float64(Float64(w * r) * r) * w), 1.5));
	else
		tmp = Float64(3.0 - fma(fma(v, -2.0, 3.0), Float64(0.125 * Float64(Float64(Float64(w * r) * w) * Float64(r / Float64(1.0 - v)))), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 35000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(0.125 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.5e7

   1. Initial program 83.1%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 93.6%

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

   if 3.5e7 < r

   1. Initial program 89.2%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* 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. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

   8. Applied rewrites 99.6%

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

Alternative 3: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+272}:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+74}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(t\_2, 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 -2e+272)
     (- (- 3.0 (* t_2 0.25)) 4.5)
     (if (<= t_1 4e+74) (- t_0 (fma t_2 0.375 1.5)) t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -2e+272) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_1 <= 4e+74) {
		tmp = t_0 - fma(t_2, 0.375, 1.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + 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)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= -2e+272)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.25)) - 4.5);
	elseif (t_1 <= 4e+74)
		tmp = Float64(t_0 - fma(t_2, 0.375, 1.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+272], N[(N[(3.0 - N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 4e+74], N[(t$95$0 - N[(t$95$2 * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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)) < -2.0000000000000001e272

   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. Taylor expanded in r around inf

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

   4. Applied rewrites 83.7%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.7

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow-prod-down N/A

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

       13. *-commutative N/A

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

       14. pow2 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 90.3

           \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5 \]

   11. Applied rewrites 90.3%

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

   if -2.0000000000000001e272 < (-.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)) < 3.99999999999999981e74

   1. Initial program 88.6%

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

   3. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 86.1%

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

   if 3.99999999999999981e74 < (-.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 82.6%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 4: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+272}:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 -2e+272)
     (- (- 3.0 (* t_2 0.25)) 4.5)
     (if (<= t_1 -1.5)
       (- (- 3.0 (* t_2 0.375)) 4.5)
       (- t_0 (fma (* (* (* r r) 0.375) w) w 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -2e+272) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	} else {
		tmp = t_0 - fma((((r * r) * 0.375) * w), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + 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)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= -2e+272)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.25)) - 4.5);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.375)) - 4.5);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.375) * w), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+272], N[(N[(3.0 - N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(t$95$2 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -2.0000000000000001e272

   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. Taylor expanded in r around inf

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

   4. Applied rewrites 83.7%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.7

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow-prod-down N/A

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

       13. *-commutative N/A

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

       14. pow2 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 90.3

           \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5 \]

   11. Applied rewrites 90.3%

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

   if -2.0000000000000001e272 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.4%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 88.7

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 83.6

          \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.375\right) - 4.5 \]

   12. Applied rewrites 83.6%

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

   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 84.1%

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

   2. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   3. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, \color{blue}{w}, 1.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+272}:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 -2e+272)
     (- (- 3.0 (* t_2 0.25)) 4.5)
     (if (<= t_1 -1.5) (- (- 3.0 (* t_2 0.375)) 4.5) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -2e+272) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	} else {
		tmp = t_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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
    t_2 = ((w * r) * w) * r
    if (t_1 <= (-2d+272)) then
        tmp = (3.0d0 - (t_2 * 0.25d0)) - 4.5d0
    else if (t_1 <= (-1.5d0)) then
        tmp = (3.0d0 - (t_2 * 0.375d0)) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -2e+272) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	t_2 = ((w * r) * w) * r
	tmp = 0
	if t_1 <= -2e+272:
		tmp = (3.0 - (t_2 * 0.25)) - 4.5
	elif t_1 <= -1.5:
		tmp = (3.0 - (t_2 * 0.375)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + 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)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= -2e+272)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.25)) - 4.5);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.375)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	t_2 = ((w * r) * w) * r;
	tmp = 0.0;
	if (t_1 <= -2e+272)
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	elseif (t_1 <= -1.5)
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+272], N[(N[(3.0 - N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(t$95$2 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 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)) < -2.0000000000000001e272

   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. Taylor expanded in r around inf

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

   4. Applied rewrites 83.7%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.7

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow-prod-down N/A

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

       13. *-commutative N/A

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

       14. pow2 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 90.3

           \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5 \]

   11. Applied rewrites 90.3%

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

   if -2.0000000000000001e272 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.4%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 88.7

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 83.6

          \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.375\right) - 4.5 \]

   12. Applied rewrites 83.6%

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

   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 84.1%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 6: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.375, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -2e+256)
     (- (- 3.0 (* (* (* (* w r) w) r) 0.25)) 4.5)
     (- t_0 (fma (* (* (* w r) r) w) 0.375 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -2e+256) {
		tmp = (3.0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else {
		tmp = t_0 - fma((((w * r) * r) * w), 0.375, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + 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) <= -2e+256)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.375, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $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+256], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -2.0000000000000001e256

   1. Initial program 83.9%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 83.9%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 89.0

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

   9. Applied rewrites 89.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow-prod-down N/A

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

       13. *-commutative N/A

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

       14. pow2 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 89.8

           \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5 \]

   11. Applied rewrites 89.8%

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

   if -2.0000000000000001e256 < (-.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 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   3. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   6. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.375, 1.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 -2e+256)
     (- (- 3.0 (* (* (* (* w r) w) r) 0.25)) 4.5)
     (if (<= t_1 -1.5)
       (- (- 3.0 (* (* (* (* w r) r) w) 0.375)) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -2e+256) {
		tmp = (3.0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.375)) - 4.5;
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
    if (t_1 <= (-2d+256)) then
        tmp = (3.0d0 - ((((w * r) * w) * r) * 0.25d0)) - 4.5d0
    else if (t_1 <= (-1.5d0)) then
        tmp = (3.0d0 - ((((w * r) * r) * w) * 0.375d0)) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -2e+256) {
		tmp = (3.0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.375)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -2e+256:
		tmp = (3.0 - ((((w * r) * w) * r) * 0.25)) - 4.5
	elif t_1 <= -1.5:
		tmp = (3.0 - ((((w * r) * r) * w) * 0.375)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + 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 <= -2e+256)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * r) * w) * 0.375)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -2e+256)
		tmp = (3.0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	elseif (t_1 <= -1.5)
		tmp = (3.0 - ((((w * r) * r) * w) * 0.375)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+256], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 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)) < -2.0000000000000001e256

   1. Initial program 83.9%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 83.9%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 89.0

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

   9. Applied rewrites 89.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow-prod-down N/A

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

       13. *-commutative N/A

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

       14. pow2 N/A

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

       15. associate-*l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 89.8

           \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5 \]

   11. Applied rewrites 89.8%

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

   if -2.0000000000000001e256 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 86.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.2%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 89.0

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

   6. Applied rewrites 89.0%

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

   7. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 77.6

          \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.375\right) - 4.5 \]

   9. Applied rewrites 77.6%

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

   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 84.1%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 8: 91.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * r) * r) * w)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(Float64(3.0 + t_1) - 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_2 <= Float64(-Inf))
		tmp = Float64(Float64(3.0 - Float64(0.25 * t_0)) - 4.5);
	elseif (t_2 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(t_0 * 0.375)) - 4.5);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

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

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(3.0 - N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, -1.5], N[(N[(3.0 - N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - 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 83.3%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 83.3%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.3

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 85.5

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

   6. Applied rewrites 85.5%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 91.5

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

   9. Applied rewrites 91.5%

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

   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 87.2%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.9%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 76.6

          \[\leadsto \left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.375\right) - 4.5 \]

   9. Applied rewrites 76.6%

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

   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 84.1%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 9: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot r\right) \cdot r\right) \cdot w\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(3 - 0.25 \cdot t\_0\right) - 4.5\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+82}:\\ \;\;\;\;-0.375 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w r) r) w))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (-
          (-
           (+ 3.0 t_1)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (- (- 3.0 (* 0.25 t_0)) 4.5)
     (if (<= t_2 -5e+82) (* -0.375 t_0) (- t_1 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * r) * r) * w;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (3.0 - (0.25 * t_0)) - 4.5;
	} else if (t_2 <= -5e+82) {
		tmp = -0.375 * t_0;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * r) * r) * w;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - (0.25 * t_0)) - 4.5;
	} else if (t_2 <= -5e+82) {
		tmp = -0.375 * t_0;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * r) * r) * w
	t_1 = 2.0 / (r * r)
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (3.0 - (0.25 * t_0)) - 4.5
	elif t_2 <= -5e+82:
		tmp = -0.375 * t_0
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * r) * r) * w)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(Float64(3.0 + t_1) - 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_2 <= Float64(-Inf))
		tmp = Float64(Float64(3.0 - Float64(0.25 * t_0)) - 4.5);
	elseif (t_2 <= -5e+82)
		tmp = Float64(-0.375 * t_0);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * r) * r) * w;
	t_1 = 2.0 / (r * r);
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (3.0 - (0.25 * t_0)) - 4.5;
	elseif (t_2 <= -5e+82)
		tmp = -0.375 * t_0;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(3.0 - N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, -5e+82], N[(-0.375 * t$95$0), $MachinePrecision], N[(t$95$1 - 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 83.3%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 83.3%

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

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 83.3

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 85.5

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

   6. Applied rewrites 85.5%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 91.5

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

   9. Applied rewrites 91.5%

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

   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)) < -5.00000000000000015e82

   1. Initial program 98.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   3. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 63.1

          \[\leadsto -0.375 \cdot \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \]

   7. Applied rewrites 63.1%

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

   if -5.00000000000000015e82 < (-.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 83.8%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

Alternative 10: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -5 \cdot 10^{+82}:\\ \;\;\;\;-0.375 \cdot \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -5e+82)
     (* -0.375 (* (* (* w r) r) w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+82) {
		tmp = -0.375 * (((w * r) * r) * w);
	} else {
		tmp = t_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 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-5d+82)) then
        tmp = (-0.375d0) * (((w * r) * r) * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+82) {
		tmp = -0.375 * (((w * r) * r) * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+82:
		tmp = -0.375 * (((w * r) * r) * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + 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) <= -5e+82)
		tmp = Float64(-0.375 * Float64(Float64(Float64(w * r) * r) * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+82)
		tmp = -0.375 * (((w * r) * r) * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $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], -5e+82], N[(-0.375 * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 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)) < -5.00000000000000015e82

   1. Initial program 85.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   3. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 82.7

          \[\leadsto -0.375 \cdot \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \]

   7. Applied rewrites 82.7%

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

   if -5.00000000000000015e82 < (-.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 83.8%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

Alternative 11: 57.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 57.4

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

4. Applied rewrites 57.4%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		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 <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 83.0%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 58.5

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

   4. Applied rewrites 58.5%

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

   if 1.1499999999999999 < r

   1. Initial program 89.1%

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

   2. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   3. 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 19.4

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

   4. Applied rewrites 19.4%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 26.4%

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

Alternative 13: 14.2% accurate, 41.6× 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 84.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
3. 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 53.6

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

4. Applied rewrites 53.6%

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

5. Taylor expanded in r around inf

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

7. Applied rewrites 14.2%

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

Reproduce

herbie shell --seed 2025120 
(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))