Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.3%

Time: 5.2s

Alternatives: 11

Speedup: 1.8×

• 54.1% 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: 84.9% 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -\mathsf{fma}\left(2 \cdot \left(r \cdot w\right), \left(r \cdot w\right) \cdot 0.125, 1.5 - t\_0\right)\\ \mathbf{if}\;v \leq -1.4 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot r\right) \cdot w}{1 - v}, \left(-r\right) \cdot w, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (- (fma (* 2.0 (* r w)) (* (* r w) 0.125) (- 1.5 t_0)))))
   (if (<= v -1.4e+177)
     t_1
     (if (<= v 5e-29)
       (fma
        (/ (* (* (* 0.125 (fma v -2.0 3.0)) r) w) (- 1.0 v))
        (* (- r) w)
        (- t_0 1.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = -fma((2.0 * (r * w)), ((r * w) * 0.125), (1.5 - t_0));
	double tmp;
	if (v <= -1.4e+177) {
		tmp = t_1;
	} else if (v <= 5e-29) {
		tmp = fma(((((0.125 * fma(v, -2.0, 3.0)) * r) * w) / (1.0 - v)), (-r * w), (t_0 - 1.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(-fma(Float64(2.0 * Float64(r * w)), Float64(Float64(r * w) * 0.125), Float64(1.5 - t_0)))
	tmp = 0.0
	if (v <= -1.4e+177)
		tmp = t_1;
	elseif (v <= 5e-29)
		tmp = fma(Float64(Float64(Float64(Float64(0.125 * fma(v, -2.0, 3.0)) * r) * w) / Float64(1.0 - v)), Float64(Float64(-r) * w), Float64(t_0 - 1.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[(2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.125), $MachinePrecision] + N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[v, -1.4e+177], t$95$1, If[LessEqual[v, 5e-29], N[(N[(N[(N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[((-r) * w), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.40000000000000001e177 or 4.99999999999999986e-29 < v

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if -1.40000000000000001e177 < v < 4.99999999999999986e-29

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 93.9%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \mathsf{fma}\left(v, -2, 3\right), \left(r \cdot w\right) \cdot 0.125, 1.5 - \frac{2}{r \cdot r}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (fma
   (* (* (/ r (- 1.0 v)) w) (fma v -2.0 3.0))
   (* (* r w) 0.125)
   (- 1.5 (/ 2.0 (* r r))))))

C

double code(double v, double w, double r) {
	return -fma((((r / (1.0 - v)) * w) * fma(v, -2.0, 3.0)), ((r * w) * 0.125), (1.5 - (2.0 / (r * r))));
}

Julia

function code(v, w, r)
	return Float64(-fma(Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * fma(v, -2.0, 3.0)), Float64(Float64(r * w) * 0.125), Float64(1.5 - Float64(2.0 / Float64(r * r)))))
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

3. Applied rewrites 90.1%

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

5. Applied rewrites 97.6%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 5 \cdot 10^{+25}:\\ \;\;\;\;-\mathsf{fma}\left(w \cdot r, \left(\left(w \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot 0.125\right) \cdot \frac{r}{1 - v}, 1.5 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\frac{-0.125}{v - 1} \cdot \left(\left(\left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot r\right) \cdot w\right), w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 5e+25)
     (-
      (fma
       (* w r)
       (* (* (* w (fma -2.0 v 3.0)) 0.125) (/ r (- 1.0 v)))
       (- 1.5 t_0)))
     (-
      t_0
      (fma
       (* (/ -0.125 (- v 1.0)) (* (* (* r (fma v -2.0 3.0)) r) w))
       w
       1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 5e+25) {
		tmp = -fma((w * r), (((w * fma(-2.0, v, 3.0)) * 0.125) * (r / (1.0 - v))), (1.5 - t_0));
	} else {
		tmp = t_0 - fma(((-0.125 / (v - 1.0)) * (((r * fma(v, -2.0, 3.0)) * r) * w)), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 5e+25)
		tmp = Float64(-fma(Float64(w * r), Float64(Float64(Float64(w * fma(-2.0, v, 3.0)) * 0.125) * Float64(r / Float64(1.0 - v))), Float64(1.5 - t_0)));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(-0.125 / Float64(v - 1.0)) * Float64(Float64(Float64(r * fma(v, -2.0, 3.0)) * r) * w)), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 5e+25], (-N[(N[(w * r), $MachinePrecision] * N[(N[(N[(w * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), N[(t$95$0 - N[(N[(N[(-0.125 / N[(v - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(r * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 5.00000000000000024e25

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 94.6%

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

   if 5.00000000000000024e25 < w

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.2

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

   3. Applied rewrites 95.2%

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

   5. Applied rewrites 77.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 87.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.6

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 87.6

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

   7. Applied rewrites 87.6%

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

Alternative 4: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1.5 - \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot 0.125\\ \mathbf{if}\;v \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;-\mathsf{fma}\left(2 \cdot \left(r \cdot w\right), t\_1, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(3 \cdot \left(r \cdot w\right), t\_1, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- 1.5 (/ 2.0 (* r r)))) (t_1 (* (* r w) 0.125)))
   (if (<= v -1.8e+17)
     (- (fma (* 2.0 (* r w)) t_1 t_0))
     (- (fma (* 3.0 (* r w)) t_1 t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 1.5 - (2.0 / (r * r));
	double t_1 = (r * w) * 0.125;
	double tmp;
	if (v <= -1.8e+17) {
		tmp = -fma((2.0 * (r * w)), t_1, t_0);
	} else {
		tmp = -fma((3.0 * (r * w)), t_1, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(1.5 - Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(r * w) * 0.125)
	tmp = 0.0
	if (v <= -1.8e+17)
		tmp = Float64(-fma(Float64(2.0 * Float64(r * w)), t_1, t_0));
	else
		tmp = Float64(-fma(Float64(3.0 * Float64(r * w)), t_1, t_0));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(1.5 - N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * 0.125), $MachinePrecision]}, If[LessEqual[v, -1.8e+17], (-N[(N[(2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), (-N[(N[(3.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.8e17

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if -1.8e17 < v

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 93.4

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

   10. Applied rewrites 93.4%

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

Alternative 5: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -\mathsf{fma}\left(2 \cdot \left(r \cdot w\right), \left(r \cdot w\right) \cdot 0.125, 1.5 - t\_0\right)\\ \mathbf{if}\;v \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-29}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot w\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (- (fma (* 2.0 (* r w)) (* (* r w) 0.125) (- 1.5 t_0)))))
   (if (<= v -1.8e+17)
     t_1
     (if (<= v 5e-29) (- t_0 (fma (* (* 0.375 (* r w)) r) w 1.5)) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = -fma((2.0 * (r * w)), ((r * w) * 0.125), (1.5 - t_0));
	double tmp;
	if (v <= -1.8e+17) {
		tmp = t_1;
	} else if (v <= 5e-29) {
		tmp = t_0 - fma(((0.375 * (r * w)) * r), w, 1.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(-fma(Float64(2.0 * Float64(r * w)), Float64(Float64(r * w) * 0.125), Float64(1.5 - t_0)))
	tmp = 0.0
	if (v <= -1.8e+17)
		tmp = t_1;
	elseif (v <= 5e-29)
		tmp = Float64(t_0 - fma(Float64(Float64(0.375 * Float64(r * w)) * r), w, 1.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[(2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.125), $MachinePrecision] + N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[v, -1.8e+17], t$95$1, If[LessEqual[v, 5e-29], N[(t$95$0 - N[(N[(N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.8e17 or 4.99999999999999986e-29 < v

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if -1.8e17 < v < 4.99999999999999986e-29

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

Alternative 6: 94.3% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -2e+17)
     (- t_0 (fma (* (* 0.25 (* r w)) r) w 1.5))
     (- t_0 (fma (* (* 0.375 (* r w)) r) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e+17) {
		tmp = t_0 - fma(((0.25 * (r * w)) * r), w, 1.5);
	} else {
		tmp = t_0 - fma(((0.375 * (r * w)) * r), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2e+17)
		tmp = Float64(t_0 - fma(Float64(Float64(0.25 * Float64(r * w)) * r), w, 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(0.375 * Float64(r * w)) * r), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e+17], N[(t$95$0 - N[(N[(N[(0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e17

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 91.7

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

   10. Applied rewrites 91.7%

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

   if -2e17 < v

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

Alternative 7: 91.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

3. Applied rewrites 90.1%

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 92.3%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f64 N/A

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

   2. lift-*.f64 91.7

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

10. Applied rewrites 91.7%

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

Alternative 8: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 3.6 \cdot 10^{-173}:\\ \;\;\;\;t\_0 - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.375 \cdot \left(\left(r \cdot r\right) \cdot w\right), w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 3.6e-173)
     (- t_0 1.5)
     (- t_0 (fma (* 0.375 (* (* r r) w)) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 3.6e-173) {
		tmp = t_0 - 1.5;
	} else {
		tmp = t_0 - fma((0.375 * ((r * r) * w)), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 3.6e-173)
		tmp = Float64(t_0 - 1.5);
	else
		tmp = Float64(t_0 - fma(Float64(0.375 * Float64(Float64(r * r) * w)), w, 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 3.6e-173], N[(t$95$0 - 1.5), $MachinePrecision], N[(t$95$0 - N[(N[(0.375 * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 3.59999999999999972e-173

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 57.5%

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

   if 3.59999999999999972e-173 < w

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 87.5

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

   10. Applied rewrites 87.5%

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

Alternative 9: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5000000000019464:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(r \cdot r\right) \cdot \left(w \cdot w\right), t\_0\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5000000000019464)
     (- (fma -0.25 (* (* r r) (* w w)) t_0) 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) <= -1.5000000000019464) {
		tmp = fma(-0.25, ((r * r) * (w * w)), t_0) - 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) <= -1.5000000000019464)
		tmp = Float64(fma(-0.25, Float64(Float64(r * r) * Float64(w * w)), t_0) - 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], -1.5000000000019464], N[(N[(-0.25 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - 1.5), $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)) < -1.5000000000019464

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. pow2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 78.5

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

   10. Applied rewrites 78.5%

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

   if -1.5000000000019464 < (-.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 84.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 90.1%

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 57.5%

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

Alternative 10: 57.5% 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 84.9%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

3. Applied rewrites 90.1%

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 92.3%

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

8. Taylor expanded in w around 0

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

10. Applied rewrites 57.5%

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

Alternative 11: 45.1% 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 84.9%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

3. Applied rewrites 90.1%

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

5. Applied rewrites 97.6%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
9. 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 45.1

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

10. Applied rewrites 45.1%

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

Reproduce

herbie shell --seed 2025140 
(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))