Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 98.4%

Time: 5.3s

Alternatives: 13

Speedup: 1.8×

• 52.8% 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: 84.3% 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: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2
         (- (- (+ 3.0 t_0) (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (/ (/ 2.0 r) r) (fma 0.25 (* (* w r) (* w r)) 1.5))
     (if (<= t_2 -1.0)
       (- (- 3.0 (/ (* t_1 (* (* w (* w r)) r)) (- 1.0 v))) 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 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((2.0 / r) / r) - fma(0.25, ((w * r) * (w * r)), 1.5);
	} else if (t_2 <= -1.0) {
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / (1.0 - v))) - 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(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(2.0 / r) / r) - fma(0.25, Float64(Float64(w * r) * Float64(w * r)), 1.5));
	elseif (t_2 <= -1.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_1 * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(t_0 - 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[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(t$95$1 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1.0], N[(N[(3.0 - N[(N[(t$95$1 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(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)) < -inf.0

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   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

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 98.0%

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

   if -1 < (-.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 86.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 99.7%

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

Alternative 2: 98.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 3: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2
         (- (- (+ 3.0 t_0) (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- (/ (/ 2.0 r) r) (fma 0.25 (* (* w r) (* w r)) 1.5))
     (if (<= t_2 -1.0)
       (- (- 3.0 (/ (* t_1 (* (* r w) (* r w))) (- 1.0 v))) 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 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((2.0 / r) / r) - fma(0.25, ((w * r) * (w * r)), 1.5);
	} else if (t_2 <= -1.0) {
		tmp = (3.0 - ((t_1 * ((r * w) * (r * w))) / (1.0 - v))) - 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(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(2.0 / r) / r) - fma(0.25, Float64(Float64(w * r) * Float64(w * r)), 1.5));
	elseif (t_2 <= -1.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_1 * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(t_0 - 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[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(t$95$1 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1.0], N[(N[(3.0 - N[(N[(t$95$1 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $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)) < -inf.0

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   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

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 97.9%

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

   if -1 < (-.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 86.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 99.7%

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

Alternative 4: 98.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   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. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

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

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

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   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

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 97.8

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

   9. Applied rewrites 97.8%

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

   if -1 < (-.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 86.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 99.7%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   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. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 7: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. 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)) < -inf.0

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   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. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

Alternative 8: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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)
        -1e+161)
     (/ (/ (fma (* r r) -1.5 2.0) r) r)
     (- 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) <= -1e+161) {
		tmp = (fma((r * r), -1.5, 2.0) / r) / r;
	} 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) <= -1e+161)
		tmp = Float64(Float64(fma(Float64(r * r), -1.5, 2.0) / r) / r);
	else
		tmp = Float64(t_0 - 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], -1e+161], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5 + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $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)) < -1e161

   1. Initial program 83.6%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 5.6

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

   5. Applied rewrites 5.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

   if -1e161 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.3%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 96.1

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

   9. Applied rewrites 96.1%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 90.4%

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

Alternative 9: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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)
        -1e+129)
     (/ (/ (* (* r r) -1.5) r) r)
     (- 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) <= -1e+129) {
		tmp = (((r * r) * -1.5) / r) / r;
	} 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) <= (-1d+129)) then
        tmp = (((r * r) * (-1.5d0)) / r) / r
    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) <= -1e+129) {
		tmp = (((r * r) * -1.5) / r) / r;
	} 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) <= -1e+129:
		tmp = (((r * r) * -1.5) / r) / r
	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) <= -1e+129)
		tmp = Float64(Float64(Float64(Float64(r * r) * -1.5) / r) / r);
	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) <= -1e+129)
		tmp = (((r * r) * -1.5) / r) / r;
	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], -1e+129], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5), $MachinePrecision] / r), $MachinePrecision] / r), $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)) < -1e129

   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. Add Preprocessing

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 1.5

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

   8. Applied rewrites 1.5%

      \[\leadsto \frac{\left(r \cdot r\right) \cdot -1.5}{\color{blue}{r} \cdot r} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.8

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

   10. Applied rewrites 42.8%

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

   if -1e129 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.2%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 97.1

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 91.6%

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

Alternative 10: 93.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in v around inf

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 94.6

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

7. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. pow2 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 94.5

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

9. Applied rewrites 94.5%

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

Alternative 11: 50.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.1:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.1) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.1) {
		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 <= 1.1d0) 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 <= 1.1) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.1:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.1)
		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 <= 1.1)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.1], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1000000000000001

   1. Initial program 84.5%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 58.3%

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

   if 1.1000000000000001 < r

   1. Initial program 89.6%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 24.2%

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

Alternative 12: 57.1% accurate, 3.7× 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 85.8%

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

3. Taylor expanded in v around inf

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 94.6

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

7. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. pow2 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 94.5

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

9. Applied rewrites 94.5%

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

10. Taylor expanded in w around 0

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

12. Applied rewrites 56.7%

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

Alternative 13: 13.8% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 14.0%

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

Reproduce

herbie shell --seed 2025077 
(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))