Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.7%

Time: 4.2s

Alternatives: 12

Speedup: 1.5×

• 53.0% 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: 85.3% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), 1.5\right)\\ \mathbf{if}\;v \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 (fma 0.25 (* (* w r) (* w r)) 1.5))))
   (if (<= v -5e+31)
     t_1
     (if (<= v 1.9e-26)
       (- (+ t_0 3.0) (fma 0.375 (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma(0.25, ((w * r) * (w * r)), 1.5);
	double tmp;
	if (v <= -5e+31) {
		tmp = t_1;
	} else if (v <= 1.9e-26) {
		tmp = (t_0 + 3.0) - fma(0.375, (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(0.25, Float64(Float64(w * r) * Float64(w * r)), 1.5))
	tmp = 0.0
	if (v <= -5e+31)
		tmp = t_1;
	elseif (v <= 1.9e-26)
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	else
		tmp = t_1;
	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[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -5e+31], t$95$1, If[LessEqual[v, 1.9e-26], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -5.00000000000000027e31 or 1.90000000000000007e-26 < v

   1. Initial program 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   if -5.00000000000000027e31 < v < 1.90000000000000007e-26

   1. Initial program 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 84.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Applied rewrites 99.7%

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

Alternative 3: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), 1.5\right)\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot w, r \cdot 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 (fma 0.25 (* (* w r) (* w r)) 1.5))))
   (if (<= v -3.6e-25)
     t_1
     (if (<= v 1.9e-26) (- t_0 (fma (* (* w r) w) (* r 0.375) 1.5)) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma(0.25, ((w * r) * (w * r)), 1.5);
	double tmp;
	if (v <= -3.6e-25) {
		tmp = t_1;
	} else if (v <= 1.9e-26) {
		tmp = t_0 - fma(((w * r) * w), (r * 0.375), 1.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(0.25, Float64(Float64(w * r) * Float64(w * r)), 1.5))
	tmp = 0.0
	if (v <= -3.6e-25)
		tmp = t_1;
	elseif (v <= 1.9e-26)
		tmp = Float64(t_0 - fma(Float64(Float64(w * r) * w), Float64(r * 0.375), 1.5));
	else
		tmp = t_1;
	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[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.6e-25], t$95$1, If[LessEqual[v, 1.9e-26], N[(t$95$0 - N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * 0.375), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.5999999999999999e-25 or 1.90000000000000007e-26 < v

   1. Initial program 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   if -3.5999999999999999e-25 < v < 1.90000000000000007e-26

   1. Initial program 85.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.9

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

   4. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 91.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 91.1

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

   8. Applied rewrites 91.1%

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

Alternative 4: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r, -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 8e+75)
   (- (/ 2.0 (* r r)) (fma 0.25 (* (* w r) (* w r)) 1.5))
   (fma (* (* (* w r) w) r) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8e+75) {
		tmp = (2.0 / (r * r)) - fma(0.25, ((w * r) * (w * r)), 1.5);
	} else {
		tmp = fma((((w * r) * w) * r), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8e+75)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(0.25, Float64(Float64(w * r) * Float64(w * r)), 1.5));
	else
		tmp = fma(Float64(Float64(Float64(w * r) * w) * r), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8e+75], 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], N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.99999999999999941e75

   1. Initial program 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   if 7.99999999999999941e75 < r

   1. Initial program 85.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 43.2

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

   10. Applied rewrites 43.2%

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

   11. Taylor expanded in w around 0

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

       1. sub-flip N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. pow-prod-down N/A

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

       6. *-commutative N/A

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

       7. pow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 50.1

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

   13. Applied rewrites 50.1%

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

Alternative 5: 93.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * t_2);
	elseif (t_1 <= -1.5)
		tmp = fma(t_2, -0.375, -1.5);
	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]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(t$95$2 * -0.375 + -1.5), $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 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. pow-prod-down N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

   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)) < -1.5

   1. Initial program 85.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 43.2

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

   10. Applied rewrites 43.2%

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

   11. Taylor expanded in w around 0

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

       1. sub-flip N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. pow-prod-down N/A

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

       6. *-commutative N/A

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

       7. pow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 50.1

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

   13. Applied rewrites 50.1%

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

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

   1. Initial program 85.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

Alternative 6: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -1000000000000:\\ \;\;\;\;-0.375 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 (- INFINITY))
     (* -0.25 t_2)
     (if (<= t_1 -1000000000000.0) (* -0.375 t_2) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1000000000000.0) {
		tmp = -0.375 * t_2;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1000000000000.0) {
		tmp = -0.375 * t_2;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	t_2 = ((w * r) * w) * r
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -0.25 * t_2
	elif t_1 <= -1000000000000.0:
		tmp = -0.375 * t_2
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * t_2);
	elseif (t_1 <= -1000000000000.0)
		tmp = Float64(-0.375 * t_2);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	t_2 = ((w * r) * w) * r;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -0.25 * t_2;
	elseif (t_1 <= -1000000000000.0)
		tmp = -0.375 * t_2;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -1000000000000.0], N[(-0.375 * t$95$2), $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 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. pow-prod-down N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

   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)) < -1e12

   1. Initial program 85.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 38.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 38.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 38.6

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

   7. Applied rewrites 38.6%

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

   if -1e12 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

Alternative 7: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)\\ \mathbf{elif}\;t\_1 \leq -1000000000000:\\ \;\;\;\;-0.375 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (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))
     (* -0.25 (* (* (* w r) w) r))
     (if (<= t_1 -1000000000000.0)
       (* -0.375 (* (* w r) (* w r)))
       (- 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 = -0.25 * (((w * r) * w) * r);
	} else if (t_1 <= -1000000000000.0) {
		tmp = -0.375 * ((w * r) * (w * r));
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * (((w * r) * w) * r);
	} else if (t_1 <= -1000000000000.0) {
		tmp = -0.375 * ((w * r) * (w * r));
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -0.25 * (((w * r) * w) * r)
	elif t_1 <= -1000000000000.0:
		tmp = -0.375 * ((w * r) * (w * r))
	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(-0.25 * Float64(Float64(Float64(w * r) * w) * r));
	elseif (t_1 <= -1000000000000.0)
		tmp = Float64(-0.375 * Float64(Float64(w * r) * Float64(w * 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);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -0.25 * (((w * r) * w) * r);
	elseif (t_1 <= -1000000000000.0)
		tmp = -0.375 * ((w * r) * (w * 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]}, 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[(-0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1000000000000.0], N[(-0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $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 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. pow-prod-down N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

   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)) < -1e12

   1. Initial program 85.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 38.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 38.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 38.6

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

   7. Applied rewrites 38.6%

      \[\leadsto -0.375 \cdot \color{blue}{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 39.1

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

   9. Applied rewrites 39.1%

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

   if -1e12 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

Alternative 8: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1000000000000:\\ \;\;\;\;-0.25 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1000000000000.0)
     (* -0.25 (* (* (* w r) w) 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) <= -1000000000000.0) {
		tmp = -0.25 * (((w * r) * w) * 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) <= (-1000000000000.0d0)) then
        tmp = (-0.25d0) * (((w * r) * w) * 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) <= -1000000000000.0) {
		tmp = -0.25 * (((w * r) * w) * 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) <= -1000000000000.0:
		tmp = -0.25 * (((w * r) * w) * 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) <= -1000000000000.0)
		tmp = Float64(-0.25 * Float64(Float64(Float64(w * r) * w) * 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) <= -1000000000000.0)
		tmp = -0.25 * (((w * r) * w) * 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], -1000000000000.0], N[(-0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1e12

   1. Initial program 85.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   5. 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. pow-prod-down N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

   if -1e12 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

Alternative 9: 56.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) - 1.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. mult-flip-rev N/A

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

   3. pow2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 56.8

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

4. Applied rewrites 56.8%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
5. Add Preprocessing

Alternative 10: 50.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ (/ 2.0 r) r) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = (2.0d0 / r) / r
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 85.3%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 43.6

         \[\leadsto \frac{\frac{2}{r}}{r} \]

   6. Applied rewrites 43.6%

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

   if 1.1499999999999999 < r

   1. Initial program 85.3%

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

   2. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 14.2%

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

Alternative 11: 50.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 85.3%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

   if 1.1499999999999999 < r

   1. Initial program 85.3%

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

   2. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 14.2%

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

Alternative 12: 14.2% accurate, 41.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in r around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

4. Applied rewrites 45.3%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 14.2%

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

Reproduce

herbie shell --seed 2025134 
(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))