Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.8%

Time: 5.6s

Alternatives: 15

Speedup: 1.2×

• 53.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ (/ 2.0 (* r r)) 3.0)
  (fma (fma -0.25 v 0.375) (* (* w r) (/ (* w r) (- 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), ((w * r) * ((w * r) / (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(w * r) * Float64(Float64(w * r) / 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[(w * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.3%

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

3. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ t_1 := t\_0 - \mathsf{fma}\left(-0.25 \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{-v}, 4.5\right)\\ \mathbf{if}\;v \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 0.0075:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.375, \left(w \cdot r\right) \cdot \frac{w \cdot r}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) 3.0))
        (t_1 (- t_0 (fma (* -0.25 v) (/ (* (* r w) (* r w)) (- v)) 4.5))))
   (if (<= v -0.5)
     t_1
     (if (<= v 0.0075)
       (- t_0 (fma 0.375 (* (* w r) (/ (* w r) (- 1.0 v))) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double t_1 = t_0 - fma((-0.25 * v), (((r * w) * (r * w)) / -v), 4.5);
	double tmp;
	if (v <= -0.5) {
		tmp = t_1;
	} else if (v <= 0.0075) {
		tmp = t_0 - fma(0.375, ((w * r) * ((w * r) / (1.0 - v))), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	t_1 = Float64(t_0 - fma(Float64(-0.25 * v), Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(-v)), 4.5))
	tmp = 0.0
	if (v <= -0.5)
		tmp = t_1;
	elseif (v <= 0.0075)
		tmp = Float64(t_0 - fma(0.375, Float64(Float64(w * r) * Float64(Float64(w * r) / Float64(1.0 - v))), 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / (-v)), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -0.5], t$95$1, If[LessEqual[v, 0.0075], N[(t$95$0 - N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -0.5 or 0.0074999999999999997 < v

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 85.0

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

   6. Applied rewrites 85.0%

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

   7. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.0

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

   9. Applied rewrites 92.0%

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

   if -0.5 < v < 0.0074999999999999997

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 83.2%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \mathbf{if}\;v \leq -6 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 0.0075:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, \left(w \cdot r\right) \cdot \frac{w \cdot r}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))
   (if (<= v -6e+60)
     t_1
     (if (<= v 0.0075)
       (- (+ t_0 3.0) (fma 0.375 (* (* w r) (/ (* w r) (- 1.0 v))) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double tmp;
	if (v <= -6e+60) {
		tmp = t_1;
	} else if (v <= 0.0075) {
		tmp = (t_0 + 3.0) - fma(0.375, ((w * r) * ((w * r) / (1.0 - v))), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	tmp = 0.0
	if (v <= -6e+60)
		tmp = t_1;
	elseif (v <= 0.0075)
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, Float64(Float64(w * r) * Float64(Float64(w * r) / Float64(1.0 - v))), 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -6e+60], t$95$1, If[LessEqual[v, 0.0075], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -5.9999999999999997e60 or 0.0074999999999999997 < v

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

   if -5.9999999999999997e60 < v < 0.0074999999999999997

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 83.2%

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

Alternative 4: 95.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -500000000:\\ \;\;\;\;\frac{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (if (<= t_1 -500000000.0)
       (* (/ (* (* (* (* w r) w) r) (fma v -2.0 3.0)) (- 1.0 v)) -0.125)
       (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else if (t_1 <= -500000000.0) {
		tmp = (((((w * r) * w) * r) * fma(v, -2.0, 3.0)) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	elseif (t_1 <= -500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(w * r) * w) * r) * fma(v, -2.0, 3.0)) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000000.0], N[(N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   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)) < -5e8

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in w around inf

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

   6. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 38.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 39.5

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 39.5

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

   10. Applied rewrites 39.5%

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

   if -5e8 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

Alternative 5: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot r\right) \cdot w\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -500000000:\\ \;\;\;\;\frac{t\_2 \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(t\_2, 0.25, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) r) w)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (if (<= t_1 -500000000.0)
       (* (/ (* t_2 (fma v -2.0 3.0)) (- 1.0 v)) -0.125)
       (- t_0 (fma t_2 0.25 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * r) * w;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else if (t_1 <= -500000000.0) {
		tmp = ((t_2 * fma(v, -2.0, 3.0)) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - fma(t_2, 0.25, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * r) * w)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	elseif (t_1 <= -500000000.0)
		tmp = Float64(Float64(Float64(t_2 * fma(v, -2.0, 3.0)) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - fma(t_2, 0.25, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000000.0], N[(N[(N[(t$95$2 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - N[(t$95$2 * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   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)) < -5e8

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in w around inf

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

   6. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 38.3%

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

   if -5e8 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

Alternative 6: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.4 \cdot 10^{-55}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.4e-55)
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (- (- (+ 3.0 t_0) (/ (* 0.375 (* (* w (* w r)) r)) 1.0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.4e-55) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else {
		tmp = ((3.0 + t_0) - ((0.375 * ((w * (w * r)) * r)) / 1.0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.4e-55)
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(w * Float64(w * r)) * r)) / 1.0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.4e-55], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.39999999999999991e-55

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   if 2.39999999999999991e-55 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 76.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 83.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.0

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 90.5%

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

Alternative 7: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -500000000:\\ \;\;\;\;\frac{\left(\left(w \cdot w\right) \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot \left(r \cdot r\right)}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (if (<= t_1 -500000000.0)
       (* (/ (* (* (* w w) (fma -2.0 v 3.0)) (* r r)) (- 1.0 v)) -0.125)
       (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else if (t_1 <= -500000000.0) {
		tmp = ((((w * w) * fma(-2.0, v, 3.0)) * (r * r)) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	elseif (t_1 <= -500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(w * w) * fma(-2.0, v, 3.0)) * Float64(r * r)) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000000.0], N[(N[(N[(N[(N[(w * w), $MachinePrecision] * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   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)) < -5e8

   1. Initial program 84.3%

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

   2. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 34.7%

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

   if -5e8 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

Alternative 8: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot r\right) \cdot w\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\frac{t\_2 \cdot 3}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(t\_2, 0.25, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) r) w)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (if (<= t_1 -2e+80)
       (* (/ (* t_2 3.0) (- 1.0 v)) -0.125)
       (- t_0 (fma t_2 0.25 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * r) * w;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else if (t_1 <= -2e+80) {
		tmp = ((t_2 * 3.0) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - fma(t_2, 0.25, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * r) * w)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	elseif (t_1 <= -2e+80)
		tmp = Float64(Float64(Float64(t_2 * 3.0) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - fma(t_2, 0.25, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+80], N[(N[(N[(t$95$2 * 3.0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - N[(t$95$2 * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   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)) < -2e80

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in w around inf

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

   6. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 38.3%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 28.5%

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

   if -2e80 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

Alternative 9: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -500000000:\\ \;\;\;\;\frac{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 3}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* (* r r) 0.25) w) w 1.5))
     (if (<= t_1 -500000000.0)
       (* (/ (* (* (* (* w r) r) w) 3.0) (- 1.0 v)) -0.125)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma((((r * r) * 0.25) * w), w, 1.5);
	} else if (t_1 <= -500000000.0) {
		tmp = (((((w * r) * r) * w) * 3.0) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(r * r) * 0.25) * w), w, 1.5));
	elseif (t_1 <= -500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(w * r) * r) * w) * 3.0) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000000.0], N[(N[(N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 3.0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      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 87.4

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

   6. Applied rewrites 87.4%

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

   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)) < -5e8

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in w around inf

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

   6. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 38.3%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 28.5%

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

   if -5e8 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 57.6

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

   4. Applied rewrites 57.6%

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

Alternative 10: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5000000020668205:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.375 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5000000020668205)
     (- t_0 (fma (* 0.375 (* r r)) (* w w) 1.5))
     (- 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) <= -1.5000000020668205) {
		tmp = t_0 - fma((0.375 * (r * r)), (w * w), 1.5);
	} 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) <= -1.5000000020668205)
		tmp = Float64(t_0 - fma(Float64(0.375 * Float64(r * r)), Float64(w * w), 1.5));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, 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], -1.5000000020668205], N[(t$95$0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision] + 1.5), $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)) < -1.5000000020668205

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -1.5000000020668205 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 57.6

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

   4. Applied rewrites 57.6%

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

Alternative 11: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -500000000:\\ \;\;\;\;\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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)
        -500000000.0)
     (* (* -0.375 (* 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) <= -500000000.0) {
		tmp = (-0.375 * (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) <= (-500000000.0d0)) then
        tmp = ((-0.375d0) * (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) <= -500000000.0) {
		tmp = (-0.375 * (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) <= -500000000.0:
		tmp = (-0.375 * (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) <= -500000000.0)
		tmp = Float64(Float64(-0.375 * 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) <= -500000000.0)
		tmp = (-0.375 * (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], -500000000.0], N[(N[(-0.375 * 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)) < -5e8

   1. Initial program 84.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in w around inf

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 33.6

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

   9. Applied rewrites 33.6%

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

   if -5e8 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 57.6

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

   4. Applied rewrites 57.6%

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

Alternative 12: 57.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. mult-flip-rev N/A

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

   3. pow2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 57.6

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

4. Applied rewrites 57.6%

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

Alternative 13: 50.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.0) (/ (/ 2.0 r) r) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.0:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.0)
		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 <= 1.0)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if r < 1

   1. Initial program 84.3%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 44.2

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

   6. Applied rewrites 44.2%

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

   if 1 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   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. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 14.2%

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

Alternative 14: 50.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.0:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.0)
		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 <= 1.0)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if r < 1

   1. Initial program 84.3%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   if 1 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 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. mult-flip-rev N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   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. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 14.2%

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

Alternative 15: 14.2% accurate, 41.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in v around 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. mult-flip-rev N/A

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

   3. pow2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 78.3

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

4. Applied rewrites 78.3%

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

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. mult-flip-rev N/A

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

   10. lower-/.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 42.4

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

7. Applied rewrites 42.4%

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

8. Taylor expanded in w around 0

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

10. Applied rewrites 14.2%

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

Reproduce

herbie shell --seed 2025136 
(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))