Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 98.6%

Time: 4.7s

Alternatives: 10

Speedup: 1.5×

• 47.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 84.7% accurate, 1.0× speedup

Math

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

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: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \left(0.25 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \mathbf{if}\;v \leq -4.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-24}:\\ \;\;\;\;\left(t\_0 - \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (- (- t_0 (* (* 0.25 (* r w)) (* w r))) 4.5)))
   (if (<= v -4.2e+61)
     t_1
     (if (<= v 8e-24) (- (- t_0 (* (* 0.375 (* r w)) (* w r))) 4.5) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((0.25 * (r * w)) * (w * r))) - 4.5;
	double tmp;
	if (v <= -4.2e+61) {
		tmp = t_1;
	} else if (v <= 8e-24) {
		tmp = (t_0 - ((0.375 * (r * w)) * (w * r))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    t_1 = (t_0 - ((0.25d0 * (r * w)) * (w * r))) - 4.5d0
    if (v <= (-4.2d+61)) then
        tmp = t_1
    else if (v <= 8d-24) then
        tmp = (t_0 - ((0.375d0 * (r * w)) * (w * r))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((0.25 * (r * w)) * (w * r))) - 4.5;
	double tmp;
	if (v <= -4.2e+61) {
		tmp = t_1;
	} else if (v <= 8e-24) {
		tmp = (t_0 - ((0.375 * (r * w)) * (w * r))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = (t_0 - ((0.25 * (r * w)) * (w * r))) - 4.5
	tmp = 0
	if v <= -4.2e+61:
		tmp = t_1
	elif v <= 8e-24:
		tmp = (t_0 - ((0.375 * (r * w)) * (w * r))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(Float64(0.25 * Float64(r * w)) * Float64(w * r))) - 4.5)
	tmp = 0.0
	if (v <= -4.2e+61)
		tmp = t_1;
	elseif (v <= 8e-24)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(r * w)) * Float64(w * r))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = (t_0 - ((0.25 * (r * w)) * (w * r))) - 4.5;
	tmp = 0.0;
	if (v <= -4.2e+61)
		tmp = t_1;
	elseif (v <= 8e-24)
		tmp = (t_0 - ((0.375 * (r * w)) * (w * r))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -4.2e+61], t$95$1, If[LessEqual[v, 8e-24], N[(N[(t$95$0 - N[(N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.2000000000000002e61 or 7.99999999999999939e-24 < v

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in v around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 93.5%

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

   11. Applied rewrites 93.5%

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

   if -4.2000000000000002e61 < v < 7.99999999999999939e-24

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{3}{8} \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) - 4.5 \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 93.3%

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

   11. Applied rewrites 93.3%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in v around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 93.5%

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

   11. Applied rewrites 93.5%

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

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

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   10. Applied rewrites 95.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot r\right)}{v - 1} \cdot w, r, \frac{2}{r \cdot r}\right) - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 95.1%

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

3. Applied rewrites 95.1%

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

5. Applied rewrites 97.7%

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

Alternative 4: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left|r\right|\\ \mathbf{if}\;\left|r\right| \leq 7 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(3 + \frac{2}{\left|r\right| \cdot \left|r\right|}\right) - \left(0.375 \cdot \left(\left|r\right| \cdot w\right)\right) \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \frac{\left|r\right|}{1 - v}\right) \cdot t\_0\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (fabs r))))
   (if (<= (fabs r) 7e-12)
     (-
      (-
       (+ 3.0 (/ 2.0 (* (fabs r) (fabs r))))
       (* (* 0.375 (* (fabs r) w)) t_0))
      4.5)
     (-
      (- 3.0 (* (* (* (fma -0.25 v 0.375) w) (/ (fabs r) (- 1.0 v))) t_0))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = w * fabs(r);
	double tmp;
	if (fabs(r) <= 7e-12) {
		tmp = ((3.0 + (2.0 / (fabs(r) * fabs(r)))) - ((0.375 * (fabs(r) * w)) * t_0)) - 4.5;
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * w) * (fabs(r) / (1.0 - v))) * t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * abs(r))
	tmp = 0.0
	if (abs(r) <= 7e-12)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(abs(r) * abs(r)))) - Float64(Float64(0.375 * Float64(abs(r) * w)) * t_0)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * w) * Float64(abs(r) / Float64(1.0 - v))) * t_0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 7e-12], N[(N[(N[(3.0 + N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * N[(N[Abs[r], $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(N[Abs[r], $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 7.0000000000000001e-12

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{3}{8} \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) - 4.5 \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 93.3%

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

   11. Applied rewrites 93.3%

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

   if 7.0000000000000001e-12 < r

   1. Initial program 84.7%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.7%

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

   4. Applied rewrites 84.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.5%

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 54.7%

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

Alternative 5: 96.4% accurate, 1.2× speedup

Math

\[-\mathsf{fma}\left(\left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot r, w, 1.5 - \frac{2}{r \cdot r}\right) \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (fma
   (* (* (* (/ r (- 1.0 v)) w) (fma -0.25 v 0.375)) r)
   w
   (- 1.5 (/ 2.0 (* r r))))))

C

double code(double v, double w, double r) {
	return -fma(((((r / (1.0 - v)) * w) * fma(-0.25, v, 0.375)) * r), w, (1.5 - (2.0 / (r * r))));
}

Julia

function code(v, w, r)
	return Float64(-fma(Float64(Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * fma(-0.25, v, 0.375)) * r), w, Float64(1.5 - Float64(2.0 / Float64(r * r)))))
end

Wolfram

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

Derivation

1. Initial program 84.7%

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

2. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 84.7%

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

4. Applied rewrites 84.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

6. Applied rewrites 84.8%

   \[\leadsto \color{blue}{-\left(1.5 - \left(\frac{2}{r \cdot r} - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

8. Applied rewrites 96.4%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot r, w, 1.5 - \frac{2}{r \cdot r}\right)} \]
9. Add Preprocessing

Alternative 6: 93.5% accurate, 1.5× speedup

Math

\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.25 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot r\right)\right) - 4.5 \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r r))) (* (* 0.25 (* r w)) (* w r))) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * (r * w)) * (w * r))) - 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))) - ((0.25d0 * (r * w)) * (w * r))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * (r * w)) * (w * r))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * (r * w)) * (w * r))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.25 * Float64(r * w)) * Float64(w * r))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - ((0.25 * (r * w)) * (w * r))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

2. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 84.7%

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

4. Applied rewrites 84.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. swap-sqr N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

6. Applied rewrites 89.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 94.5%

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

8. Applied rewrites 94.5%

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

9. Taylor expanded in v around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 93.5%

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

11. Applied rewrites 93.5%

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) - 4.5 \]
12. Add Preprocessing

Alternative 7: 83.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1.5)
   (- (- 3.0 (/ (* 0.375 (* (* w r) (* w r))) (- 1.0 v))) 4.5)
   (- (* 2.0 (/ 1.0 (pow r 2.0))) 1.5)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1.5d0)) then
        tmp = (3.0d0 - ((0.375d0 * ((w * r) * (w * r))) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (2.0d0 * (1.0d0 / (r ** 2.0d0))) - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 * (1.0 / Math.pow(r, 2.0))) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5:
		tmp = (3.0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5
	else:
		tmp = (2.0 * (1.0 / math.pow(r, 2.0))) - 1.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5)
		tmp = (3.0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	else
		tmp = (2.0 * (1.0 / (r ^ 2.0))) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 * N[(1.0 / N[Power[r, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 95.1%

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

   3. Applied rewrites 95.1%

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

   4. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 41.2%

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

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

   1. Initial program 84.7%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 57.8%

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

   4. Applied rewrites 57.8%

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

Alternative 8: 83.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 95.1%

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

   3. Applied rewrites 95.1%

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

   4. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 41.2%

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

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

   1. Initial program 84.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in w around 0

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

   6. Applied rewrites 57.7%

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

Alternative 9: 57.7% accurate, 3.3× speedup

Math

\[\mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, -1.5\right) \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. mult-flip N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

3. Applied rewrites 91.4%

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

4. Taylor expanded in w around 0

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

6. Applied rewrites 57.7%

   \[\leadsto \mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, \color{blue}{-1.5}\right) \]
7. Add Preprocessing

Alternative 10: 44.6% accurate, 5.7× speedup

Math

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

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 84.7%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 44.6%

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

4. Applied rewrites 44.6%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.6%

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

6. Applied rewrites 44.6%

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

Reproduce

herbie shell --seed 2025179 
(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))