Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.4%

Time: 4.8s

Alternatives: 10

Speedup: 1.6×

• 52.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} - 1.5\\ t_1 := \left(w \cdot r\right) \cdot 0.125\\ t_2 := \mathsf{fma}\left(-2 \cdot \left(r \cdot w\right), t\_1, t\_0\right)\\ \mathbf{if}\;v \leq -3.25 \cdot 10^{+41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot \left(r \cdot w\right), t\_1, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (* r r)) 1.5))
        (t_1 (* (* w r) 0.125))
        (t_2 (fma (* -2.0 (* r w)) t_1 t_0)))
   (if (<= v -3.25e+41)
     t_2
     (if (<= v 3e-19) (fma (* -3.0 (* r w)) t_1 t_0) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) - 1.5;
	double t_1 = (w * r) * 0.125;
	double t_2 = fma((-2.0 * (r * w)), t_1, t_0);
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_2;
	} else if (v <= 3e-19) {
		tmp = fma((-3.0 * (r * w)), t_1, t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) - 1.5)
	t_1 = Float64(Float64(w * r) * 0.125)
	t_2 = fma(Float64(-2.0 * Float64(r * w)), t_1, t_0)
	tmp = 0.0
	if (v <= -3.25e+41)
		tmp = t_2;
	elseif (v <= 3e-19)
		tmp = fma(Float64(-3.0 * Float64(r * w)), t_1, t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * r), $MachinePrecision] * 0.125), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]}, If[LessEqual[v, -3.25e+41], t$95$2, If[LessEqual[v, 3e-19], N[(N[(-3.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.24999999999999988e41 or 2.99999999999999993e-19 < v

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if -3.24999999999999988e41 < v < 2.99999999999999993e-19

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{-3 \cdot \left(r \cdot w\right)}, \left(w \cdot r\right) \cdot 0.125, \frac{2}{r \cdot r} - 1.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 93.1

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

3. Applied rewrites 93.1%

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

5. Applied rewrites 99.3%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 93.1

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

3. Applied rewrites 93.1%

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

5. Applied rewrites 99.4%

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

Alternative 4: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5500000:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot \left(r \cdot w\right), \left(w \cdot r\right) \cdot 0.125, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(v, 2, -3\right) \cdot \left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \left(w \cdot r\right)\right), 0.125, 3\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5500000.0)
   (fma (* -3.0 (* r w)) (* (* w r) 0.125) (- (/ 2.0 (* r r)) 1.5))
   (-
    (fma (* (fma v 2.0 -3.0) (* (* (/ r (- 1.0 v)) w) (* w r))) 0.125 3.0)
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5500000.0) {
		tmp = fma((-3.0 * (r * w)), ((w * r) * 0.125), ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = fma((fma(v, 2.0, -3.0) * (((r / (1.0 - v)) * w) * (w * r))), 0.125, 3.0) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5500000.0)
		tmp = fma(Float64(-3.0 * Float64(r * w)), Float64(Float64(w * r) * 0.125), Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = Float64(fma(Float64(fma(v, 2.0, -3.0) * Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * Float64(w * r))), 0.125, 3.0) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5500000.0], N[(N[(-3.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * 0.125), $MachinePrecision] + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.5e6

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if 5.5e6 < r

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 55.8%

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

Alternative 5: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot r\right) \cdot 0.125\\ \mathbf{if}\;r \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(r \cdot w\right), t\_0, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(2, v, -3\right) \cdot w\right) \cdot \frac{r}{1 - v}, t\_0, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w r) 0.125)))
   (if (<= r 1.6e+15)
     (fma (* -2.0 (* r w)) t_0 (- (/ 2.0 (* r r)) 1.5))
     (fma (* (* (fma 2.0 v -3.0) w) (/ r (- 1.0 v))) t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * 0.125;
	double tmp;
	if (r <= 1.6e+15) {
		tmp = fma((-2.0 * (r * w)), t_0, ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = fma(((fma(2.0, v, -3.0) * w) * (r / (1.0 - v))), t_0, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * 0.125)
	tmp = 0.0
	if (r <= 1.6e+15)
		tmp = fma(Float64(-2.0 * Float64(r * w)), t_0, Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = fma(Float64(Float64(fma(2.0, v, -3.0) * w) * Float64(r / Float64(1.0 - v))), t_0, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * 0.125), $MachinePrecision]}, If[LessEqual[r, 1.6e+15], N[(N[(-2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * v + -3.0), $MachinePrecision] * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.6e15

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if 1.6e15 < r

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 55.1%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(2, v, -3\right) \cdot w\right) \cdot \frac{r}{1 - v}, \left(w \cdot r\right) \cdot 0.125, \color{blue}{-1.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-3 \cdot \left(r \cdot w\right), \left(w \cdot r\right) \cdot 0.125, \frac{2}{r \cdot r} - 1.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (fma (* -3.0 (* r w)) (* (* w r) 0.125) (- (/ 2.0 (* r r)) 1.5)))

C

double code(double v, double w, double r) {
	return fma((-3.0 * (r * w)), ((w * r) * 0.125), ((2.0 / (r * r)) - 1.5));
}

Julia

function code(v, w, r)
	return fma(Float64(-3.0 * Float64(r * w)), Float64(Float64(w * r) * 0.125), Float64(Float64(2.0 / Float64(r * r)) - 1.5))
end

Wolfram

code[v_, w_, r_] := N[(N[(-3.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * 0.125), $MachinePrecision] + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 93.1

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

3. Applied rewrites 93.1%

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

5. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 94.8%

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

8. Taylor expanded in v around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 93.3

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

10. Applied rewrites 93.3%

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

Alternative 7: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\\ \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.505:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(v, 2, -3\right) \cdot \left(t\_0 \cdot 0.125\right), r, 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{r}^{2}} - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w w) r)))
   (if (<=
        (-
         (-
          (+ 3.0 (/ 2.0 (* r r)))
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r)) (- 1.0 v)))
         4.5)
        -1.505)
     (- (fma (* (fma v 2.0 -3.0) (* t_0 0.125)) r 3.0) 4.5)
     (- (* 2.0 (/ 1.0 (pow r 2.0))) 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = (w * w) * r;
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) - 4.5) <= -1.505) {
		tmp = fma((fma(v, 2.0, -3.0) * (t_0 * 0.125)), r, 3.0) - 4.5;
	} else {
		tmp = (2.0 * (1.0 / pow(r, 2.0))) - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * w) * r)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r)) / Float64(1.0 - v))) - 4.5) <= -1.505)
		tmp = Float64(fma(Float64(fma(v, 2.0, -3.0) * Float64(t_0 * 0.125)), r, 3.0) - 4.5);
	else
		tmp = Float64(Float64(2.0 * Float64(1.0 / (r ^ 2.0))) - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[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[(t$95$0 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.505], N[(N[(N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision] * r + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 * N[(1.0 / N[Power[r, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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)) < -1.5049999999999999

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 39.8%

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

   if -1.5049999999999999 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 57.2

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - 1.5 \]

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 39.8%

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

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

   1. Initial program 85.3%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.1

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

   6. Applied rewrites 44.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 44.1

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

   8. Applied rewrites 44.1%

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

Alternative 9: 44.1% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (v w r) :precision binary64 (/ (/ 2.0 r) r))

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 44.1

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

4. Applied rewrites 44.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.1

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

6. Applied rewrites 44.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 44.1

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

8. Applied rewrites 44.1%

   \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]
9. Add Preprocessing

Alternative 10: 44.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 44.1

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

4. Applied rewrites 44.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.1

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

6. Applied rewrites 44.1%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(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))