Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.7%

Time: 5.1s

Alternatives: 14

Speedup: 1.5×

• 53.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_1 := \frac{2}{r \cdot r} + 3\\ t_2 := t\_1 - \mathsf{fma}\left(-0.25 \cdot v, t\_0, 4.5\right)\\ \mathbf{if}\;v \leq -6.5 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(0.375, t\_0, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (* (* r w) (* r w)) (- 1.0 v)))
        (t_1 (+ (/ 2.0 (* r r)) 3.0))
        (t_2 (- t_1 (fma (* -0.25 v) t_0 4.5))))
   (if (<= v -6.5e+38) t_2 (if (<= v 1.5) (- t_1 (fma 0.375 t_0 4.5)) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) / (1.0 - v);
	double t_1 = (2.0 / (r * r)) + 3.0;
	double t_2 = t_1 - fma((-0.25 * v), t_0, 4.5);
	double tmp;
	if (v <= -6.5e+38) {
		tmp = t_2;
	} else if (v <= 1.5) {
		tmp = t_1 - fma(0.375, t_0, 4.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	t_2 = Float64(t_1 - fma(Float64(-0.25 * v), t_0, 4.5))
	tmp = 0.0
	if (v <= -6.5e+38)
		tmp = t_2;
	elseif (v <= 1.5)
		tmp = Float64(t_1 - fma(0.375, t_0, 4.5));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(-0.25 * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -6.5e+38], t$95$2, If[LessEqual[v, 1.5], N[(t$95$1 - N[(0.375 * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.5e38 or 1.5 < v

   1. Initial program 81.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -6.5e38 < v < 1.5

   1. Initial program 87.6%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

      1. +-commutative 98.5

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

      2. fp-cancel-sign-sub-inv 98.5

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

      3. metadata-eval 98.5

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

      4. *-commutative 98.5

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

   6. Applied rewrites 98.5%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Applied rewrites 99.7%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \mathbf{if}\;v \leq -6.5 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))
   (if (<= v -6.5e+38)
     t_1
     (if (<= v 1.5)
       (- (+ t_0 3.0) (fma 0.375 (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double tmp;
	if (v <= -6.5e+38) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = (t_0 + 3.0) - fma(0.375, (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	tmp = 0.0
	if (v <= -6.5e+38)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -6.5e+38], t$95$1, If[LessEqual[v, 1.5], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.5e38 or 1.5 < v

   1. Initial program 81.3%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   if -6.5e38 < v < 1.5

   1. Initial program 87.6%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

      1. +-commutative 98.5

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

      2. fp-cancel-sign-sub-inv 98.5

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

      3. metadata-eval 98.5

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

      4. *-commutative 98.5

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

   6. Applied rewrites 98.5%

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

Alternative 4: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ t_2 := \left(w \cdot w\right) \cdot r\\ t_3 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_2 \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_2\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5)))
        (t_2 (* (* w w) r))
        (t_3
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_2 r)) (- 1.0 v)))
          4.5)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -2e+45)
       (- (- 3.0 (/ (* (* (* (fma v -2.0 3.0) 0.125) t_2) r) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double t_2 = (w * w) * r;
	double t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (t_2 * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -2e+45) {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * t_2) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	t_2 = Float64(Float64(w * w) * r)
	t_3 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_2 * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -2e+45)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_2) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -2e+45], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$2), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]]]

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 or -1.9999999999999999e45 < (-.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 83.5%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

   if -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)) < -1.9999999999999999e45

   1. Initial program 98.5%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 97.8%

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

   6. Applied rewrites 97.5%

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

Alternative 5: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot r\right) \cdot w\\ t_3 := t\_0 - \mathsf{fma}\left(t\_2, 0.25, 1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_2}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) r) w))
        (t_3 (- t_0 (fma t_2 0.25 1.5))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1.5)
       (- (- 3.0 (/ (* (* (fma v -2.0 3.0) 0.125) t_2) (- 1.0 v))) 4.5)
       t_3))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * r) * w;
	double t_3 = t_0 - fma(t_2, 0.25, 1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (((fma(v, -2.0, 3.0) * 0.125) * t_2) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * r) * w)
	t_3 = Float64(t_0 - fma(t_2, 0.25, 1.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_2) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[(t$95$2 * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$3]]]]]]

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 or -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 83.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if -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)) < -1.5

   1. Initial program 87.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 6: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;-0.375 \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - 0.25 \cdot t\_2\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 (- INFINITY))
     (* -0.25 t_2)
     (if (<= t_1 -2e+45)
       (* -0.375 t_2)
       (if (<= t_1 -1.5) (- (- 3.0 (* 0.25 t_2)) 4.5) (- t_0 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -2e+45) {
		tmp = -0.375 * t_2;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (0.25 * t_2)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -2e+45) {
		tmp = -0.375 * t_2;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (0.25 * t_2)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	t_2 = ((w * r) * w) * r
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -0.25 * t_2
	elif t_1 <= -2e+45:
		tmp = -0.375 * t_2
	elif t_1 <= -1.5:
		tmp = (3.0 - (0.25 * t_2)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * t_2);
	elseif (t_1 <= -2e+45)
		tmp = Float64(-0.375 * t_2);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(0.25 * t_2)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	t_2 = ((w * r) * w) * r;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -0.25 * t_2;
	elseif (t_1 <= -2e+45)
		tmp = -0.375 * t_2;
	elseif (t_1 <= -1.5)
		tmp = (3.0 - (0.25 * t_2)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -2e+45], N[(-0.375 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(0.25 * t$95$2), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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 82.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.9999999999999999e45

   1. Initial program 98.5%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -1.9999999999999999e45 < (-.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 82.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

   10. Applied rewrites 90.3%

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

   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 84.4%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 7: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 500000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;r \leq 4.8 \cdot 10^{+111}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(v, -2, 3\right), 0.125 \cdot \left(\left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}\right), 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{1}{\frac{1}{\left(\left(w \cdot r\right) \cdot w\right) \cdot r}} \cdot 0.375\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 500000.0)
   (- (/ 2.0 (* r r)) (fma (* (* (* r r) 0.375) w) w 1.5))
   (if (<= r 4.8e+111)
     (-
      3.0
      (fma (fma v -2.0 3.0) (* 0.125 (* (* r r) (/ (* w w) (- 1.0 v)))) 4.5))
     (- (- 3.0 (* (/ 1.0 (/ 1.0 (* (* (* w r) w) r))) 0.375)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 500000.0) {
		tmp = (2.0 / (r * r)) - fma((((r * r) * 0.375) * w), w, 1.5);
	} else if (r <= 4.8e+111) {
		tmp = 3.0 - fma(fma(v, -2.0, 3.0), (0.125 * ((r * r) * ((w * w) / (1.0 - v)))), 4.5);
	} else {
		tmp = (3.0 - ((1.0 / (1.0 / (((w * r) * w) * r))) * 0.375)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 500000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(Float64(r * r) * 0.375) * w), w, 1.5));
	elseif (r <= 4.8e+111)
		tmp = Float64(3.0 - fma(fma(v, -2.0, 3.0), Float64(0.125 * Float64(Float64(r * r) * Float64(Float64(w * w) / Float64(1.0 - v)))), 4.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(w * r) * w) * r))) * 0.375)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 500000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 4.8e+111], N[(3.0 - N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(0.125 * N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(1.0 / N[(1.0 / N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5e5

   1. Initial program 83.1%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 78.7

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

   4. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 90.5

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

   6. Applied rewrites 90.5%

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

   if 5e5 < r < 4.80000000000000011e111

   1. Initial program 93.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 93.5%

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

   6. Applied rewrites 98.7%

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

   if 4.80000000000000011e111 < r

   1. Initial program 86.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 67.0%

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

   8. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 88.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. pow2 N/A

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

       6. metadata-eval N/A

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

       7. pow-unpow N/A

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

       8. *-commutative N/A

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

       9. pow-prod-down N/A

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

       10. metadata-eval N/A

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

       11. pow-flip N/A

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

       12. inv-pow N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. unpow-prod-down N/A

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

       17. pow2 N/A

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

       18. associate-*l* N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. lift-*.f64 88.3

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

   12. Applied rewrites 88.3%

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

Alternative 8: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 (- INFINITY))
     (* -0.25 t_2)
     (if (<= t_1 -1.5) (- (- 3.0 (* t_2 0.375)) 4.5) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	t_2 = ((w * r) * w) * r
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -0.25 * t_2
	elif t_1 <= -1.5:
		tmp = (3.0 - (t_2 * 0.375)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * t_2);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.375)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	t_2 = ((w * r) * w) * r;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -0.25 * t_2;
	elseif (t_1 <= -1.5)
		tmp = (3.0 - (t_2 * 0.375)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(t$95$2 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 87.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 83.3%

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

   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 84.4%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 9: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot r\right) \cdot w\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(t\_0 \cdot r\right)\\ \mathbf{elif}\;t\_2 \leq -1.5:\\ \;\;\;\;\left(3 - t\_0 \cdot \left(r \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w r) w))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (-
          (-
           (+ 3.0 t_1)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (* -0.25 (* t_0 r))
     (if (<= t_2 -1.5) (- (- 3.0 (* t_0 (* r 0.375))) 4.5) (- t_1 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * w;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * (t_0 * r);
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - (t_0 * (r * 0.375))) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = (w * r) * w;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * (t_0 * r);
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - (t_0 * (r * 0.375))) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (w * r) * w
	t_1 = 2.0 / (r * r)
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -0.25 * (t_0 * r)
	elif t_2 <= -1.5:
		tmp = (3.0 - (t_0 * (r * 0.375))) - 4.5
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * w)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(Float64(3.0 + 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)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(t_0 * r));
	elseif (t_2 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(t_0 * Float64(r * 0.375))) - 4.5);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (w * r) * w;
	t_1 = 2.0 / (r * r);
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -0.25 * (t_0 * r);
	elseif (t_2 <= -1.5)
		tmp = (3.0 - (t_0 * (r * 0.375))) - 4.5;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(t$95$0 * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1.5], N[(N[(3.0 - N[(t$95$0 * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - 1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 87.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 83.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 83.3

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

   12. Applied rewrites 83.3%

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

   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 84.4%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 10: 90.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - 0.25 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9e+43)
   (- (/ 2.0 (* r r)) (fma (* (* (* r r) 0.375) w) w 1.5))
   (- (- 3.0 (* 0.25 (* (* (* w r) w) r))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+43) {
		tmp = (2.0 / (r * r)) - fma((((r * r) * 0.375) * w), w, 1.5);
	} else {
		tmp = (3.0 - (0.25 * (((w * r) * w) * r))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9e+43)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(Float64(r * r) * 0.375) * w), w, 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(0.25 * Float64(Float64(Float64(w * r) * w) * r))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9e+43], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9e43

   1. Initial program 83.5%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.1

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

   4. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 90.4

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

   6. Applied rewrites 90.4%

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

   if 9e43 < r

   1. Initial program 88.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

   10. Applied rewrites 90.6%

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

Alternative 11: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;-0.375 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (* (* (* w r) w) r)))
   (if (<= t_1 (- INFINITY))
     (* -0.25 t_2)
     (if (<= t_1 -1e+28) (* -0.375 t_2) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1e+28) {
		tmp = -0.375 * t_2;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = ((w * r) * w) * r;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.25 * t_2;
	} else if (t_1 <= -1e+28) {
		tmp = -0.375 * t_2;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	t_2 = ((w * r) * w) * r
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -0.25 * t_2
	elif t_1 <= -1e+28:
		tmp = -0.375 * t_2
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(Float64(Float64(w * r) * w) * r)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * t_2);
	elseif (t_1 <= -1e+28)
		tmp = Float64(-0.375 * t_2);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	t_2 = ((w * r) * w) * r;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -0.25 * t_2;
	elseif (t_1 <= -1e+28)
		tmp = -0.375 * t_2;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -1e+28], N[(-0.375 * t$95$2), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -9.99999999999999958e27

   1. Initial program 98.5%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if -9.99999999999999958e27 < (-.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 83.9%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

Alternative 12: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;-0.25 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1e+28)
     (* -0.25 (* (* (* w r) w) r))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+28) {
		tmp = -0.25 * (((w * r) * w) * r);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1d+28)) then
        tmp = (-0.25d0) * (((w * r) * w) * r)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+28) {
		tmp = -0.25 * (((w * r) * w) * r);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+28:
		tmp = -0.25 * (((w * r) * w) * r)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1e+28)
		tmp = Float64(-0.25 * Float64(Float64(Float64(w * r) * w) * r));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+28)
		tmp = -0.25 * (((w * r) * w) * r);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1e+28], N[(-0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -9.99999999999999958e27

   1. Initial program 85.6%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 6.6%

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

   8. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 83.1

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

   10. Applied rewrites 83.1%

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

   if -9.99999999999999958e27 < (-.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 83.9%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

Alternative 13: 57.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 57.3

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

4. Applied rewrites 57.3%

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

Alternative 14: 44.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

2. Taylor expanded in r around 0

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

   1. pow2 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 44.3

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

4. Applied rewrites 44.3%

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

Reproduce

herbie shell --seed 2025116 
(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))