Rosa's TurbineBenchmark

Percentage Accurate: 85.5% → 99.7%

Time: 4.2s

Alternatives: 14

Speedup: 1.8×

• 52.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% 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, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ (/ 2.0 (* r r)) 3.0)
  (fma (fma -0.25 v 0.375) (/ (* (* 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(-0.25, v, 0.375), (((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(fma(-0.25, v, 0.375), 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[(-0.25 * v + 0.375), $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 85.5%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

   6. lower-fma.f64 99.7

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
7. Add Preprocessing

Alternative 2: 93.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3000000:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3000000.0)
   (- (/ (/ 2.0 r) r) (fma (* (* (* r r) 0.25) w) w 1.5))
   (- 3.0 (fma (fma -0.25 v 0.375) (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3000000.0) {
		tmp = ((2.0 / r) / r) - fma((((r * r) * 0.25) * w), w, 1.5);
	} else {
		tmp = 3.0 - fma(fma(-0.25, v, 0.375), (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3000000.0)
		tmp = Float64(Float64(Float64(2.0 / r) / r) - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	else
		tmp = Float64(3.0 - fma(fma(-0.25, v, 0.375), Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3000000.0], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3e6

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if 3e6 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

      6. lower-fma.f64 99.7

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 56.4%

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

Alternative 3: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3000000:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \left(w \cdot r\right) \cdot \frac{w \cdot r}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3000000.0)
   (- (/ (/ 2.0 r) r) (fma (* (* (* r r) 0.25) w) w 1.5))
   (- 3.0 (fma (fma -0.25 v 0.375) (* (* w r) (/ (* w r) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3000000.0) {
		tmp = ((2.0 / r) / r) - fma((((r * r) * 0.25) * w), w, 1.5);
	} else {
		tmp = 3.0 - fma(fma(-0.25, v, 0.375), ((w * r) * ((w * r) / (1.0 - v))), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3000000.0)
		tmp = Float64(Float64(Float64(2.0 / r) / r) - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	else
		tmp = Float64(3.0 - fma(fma(-0.25, v, 0.375), Float64(Float64(w * r) * Float64(Float64(w * r) / Float64(1.0 - v))), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3000000.0], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3e6

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if 3e6 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

      6. lower-fma.f64 99.7

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 56.4%

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

       1. lift--.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift--.f64 56.4

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

   11. Applied rewrites 56.4%

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

Alternative 4: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right) \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 1.2e+17)
     (- (- (+ 3.0 t_0) (* (* (* w (* w r)) r) 0.375)) 4.5)
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1.2e+17) {
		tmp = ((3.0 + t_0) - (((w * (w * r)) * r) * 0.375)) - 4.5;
	} else {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 1.2e+17)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * Float64(w * r)) * r) * 0.375)) - 4.5);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 1.2e+17], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 1.2e17

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 90.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 90.7

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

   7. Applied rewrites 90.7%

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

   if 1.2e17 < w

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

      \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, \color{blue}{w}, 1.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r, 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 1.2e+17)
     (- t_0 (fma (* (* w (* w r)) r) 0.375 1.5))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1.2e+17) {
		tmp = t_0 - fma(((w * (w * r)) * r), 0.375, 1.5);
	} else {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 1.2e+17)
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(w * r)) * r), 0.375, 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 1.2e+17], N[(t$95$0 - N[(N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 1.2e17

   1. Initial program 85.5%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-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 90.7%

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

   if 1.2e17 < w

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

      \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, \color{blue}{w}, 1.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 37.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 37.8

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

   7. Applied rewrites 37.8%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1

   1. Initial program 85.5%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

      6. lower-fma.f64 99.7

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 56.4%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 42.0%

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

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

   1. Initial program 85.5%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

Alternative 7: 90.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (- (/ 2.0 (* r r)) (fma (* (* w (* w r)) r) 0.375 1.5)))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) - fma(((w * (w * r)) * r), 0.375, 1.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(w * Float64(w * r)) * r), 0.375, 1.5))
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-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 90.7%

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

Alternative 8: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5000000000276814:\\ \;\;\;\;-\left(\mathsf{fma}\left(w \cdot w, 0.25, \frac{1.5}{r \cdot r}\right) \cdot r\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5000000000276814)
     (- (* (* (fma (* w w) 0.25 (/ 1.5 (* r r))) r) r))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5000000000276814) {
		tmp = -((fma((w * w), 0.25, (1.5 / (r * r))) * r) * r);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.5000000000276814)
		tmp = Float64(-Float64(Float64(fma(Float64(w * w), 0.25, Float64(1.5 / Float64(r * r))) * r) * r));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5000000000276814], (-N[(N[(N[(N[(w * w), $MachinePrecision] * 0.25 + N[(1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5000000000276814

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.5

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

   9. Applied rewrites 44.5%

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

   if -1.5000000000276814 < (-.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 85.5%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

Alternative 9: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -5e+47)
     (* (* (* w (* w r)) r) -0.25)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47) {
		tmp = ((w * (w * r)) * r) * -0.25;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-5d+47)) then
        tmp = ((w * (w * r)) * r) * (-0.25d0)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47) {
		tmp = ((w * (w * r)) * r) * -0.25;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47:
		tmp = ((w * (w * r)) * r) * -0.25
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -5e+47)
		tmp = Float64(Float64(Float64(w * Float64(w * r)) * r) * -0.25);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47)
		tmp = ((w * (w * r)) * r) * -0.25;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -5e+47], N[(N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * -0.25), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -5.00000000000000022e47

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 37.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 37.8

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

   7. Applied rewrites 37.8%

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

   if -5.00000000000000022e47 < (-.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 85.5%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

Alternative 10: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -5e+47)
     (* (* -0.25 (* r r)) (* w w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-5d+47)) then
        tmp = ((-0.25d0) * (r * r)) * (w * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47:
		tmp = (-0.25 * (r * r)) * (w * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -5e+47)
		tmp = Float64(Float64(-0.25 * Float64(r * r)) * Float64(w * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+47)
		tmp = (-0.25 * (r * r)) * (w * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -5e+47], N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -5.00000000000000022e47

   1. Initial program 85.5%

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

   2. Taylor expanded in v around inf

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

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 33.9

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

   10. Applied rewrites 33.9%

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

   if -5.00000000000000022e47 < (-.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 85.5%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

Alternative 11: 57.2% 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 85.5%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 57.2

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

4. Applied rewrites 57.2%

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

Alternative 12: 50.3% 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 85.5%

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

   2. 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 43.9

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   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-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 43.9

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

   6. Applied rewrites 43.9%

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

   if 1.1499999999999999 < r

   1. Initial program 85.5%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 14.3%

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

Alternative 13: 50.3% 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 85.5%

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

   2. 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 43.9

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

   4. Applied rewrites 43.9%

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

   if 1.1499999999999999 < r

   1. Initial program 85.5%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 14.3%

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

Alternative 14: 14.3% 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 85.5%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 53.6

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

4. Applied rewrites 53.6%

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

5. Taylor expanded in r around inf

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

7. Applied rewrites 14.3%

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

Reproduce

herbie shell --seed 2025130 
(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))