Rosa's TurbineBenchmark

Percentage Accurate: 85.2% → 99.7%

Time: 4.0s

Alternatives: 11

Speedup: 1.5×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 1.5 - \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot 0.125\right) \cdot r\\ t_2 := -\mathsf{fma}\left(t\_1, 2 \cdot \left(r \cdot w\right), t\_0\right)\\ \mathbf{if}\;v \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;-\mathsf{fma}\left(t\_1, 3 \cdot \left(r \cdot w\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- 1.5 (/ 2.0 (* r r))))
        (t_1 (* (* w 0.125) r))
        (t_2 (- (fma t_1 (* 2.0 (* r w)) t_0))))
   (if (<= v -2e+46)
     t_2
     (if (<= v 2.7e-13) (- (fma t_1 (* 3.0 (* r w)) t_0)) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = 1.5 - (2.0 / (r * r));
	double t_1 = (w * 0.125) * r;
	double t_2 = -fma(t_1, (2.0 * (r * w)), t_0);
	double tmp;
	if (v <= -2e+46) {
		tmp = t_2;
	} else if (v <= 2.7e-13) {
		tmp = -fma(t_1, (3.0 * (r * w)), t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(1.5 - Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(w * 0.125) * r)
	t_2 = Float64(-fma(t_1, Float64(2.0 * Float64(r * w)), t_0))
	tmp = 0.0
	if (v <= -2e+46)
		tmp = t_2;
	elseif (v <= 2.7e-13)
		tmp = Float64(-fma(t_1, Float64(3.0 * Float64(r * w)), t_0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(1.5 - N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * 0.125), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = (-N[(t$95$1 * N[(2.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision])}, If[LessEqual[v, -2e+46], t$95$2, If[LessEqual[v, 2.7e-13], (-N[(t$95$1 * N[(3.0 * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e46 or 2.7000000000000001e-13 < v

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 85.2%

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

   3. Applied rewrites 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 92.9%

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

   8. Applied rewrites 92.9%

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

   if -2e46 < v < 2.7000000000000001e-13

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 85.2%

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

   3. Applied rewrites 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 93.5%

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

   8. Applied rewrites 93.5%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{\left|r\right| \cdot \left|r\right|}\\ t_1 := \left(w \cdot 0.125\right) \cdot \left|r\right|\\ \mathbf{if}\;\left|r\right| \leq 3.895 \cdot 10^{-43}:\\ \;\;\;\;-\mathsf{fma}\left(t\_1, 2 \cdot \left(\left|r\right| \cdot w\right), 1.5 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(\mathsf{fma}\left(v, 2, -3\right) \cdot \left(\left(t\_1 \cdot w\right) \cdot \frac{\left|r\right|}{1 - v}\right) - 1.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* (fabs r) (fabs r)))) (t_1 (* (* w 0.125) (fabs r))))
   (if (<= (fabs r) 3.895e-43)
     (- (fma t_1 (* 2.0 (* (fabs r) w)) (- 1.5 t_0)))
     (+
      t_0
      (- (* (fma v 2.0 -3.0) (* (* t_1 w) (/ (fabs r) (- 1.0 v)))) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (fabs(r) * fabs(r));
	double t_1 = (w * 0.125) * fabs(r);
	double tmp;
	if (fabs(r) <= 3.895e-43) {
		tmp = -fma(t_1, (2.0 * (fabs(r) * w)), (1.5 - t_0));
	} else {
		tmp = t_0 + ((fma(v, 2.0, -3.0) * ((t_1 * w) * (fabs(r) / (1.0 - v)))) - 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(abs(r) * abs(r)))
	t_1 = Float64(Float64(w * 0.125) * abs(r))
	tmp = 0.0
	if (abs(r) <= 3.895e-43)
		tmp = Float64(-fma(t_1, Float64(2.0 * Float64(abs(r) * w)), Float64(1.5 - t_0)));
	else
		tmp = Float64(t_0 + Float64(Float64(fma(v, 2.0, -3.0) * Float64(Float64(t_1 * w) * Float64(abs(r) / Float64(1.0 - v)))) - 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * 0.125), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 3.895e-43], (-N[(t$95$1 * N[(2.0 * N[(N[Abs[r], $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] + N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), N[(t$95$0 + N[(N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(N[(t$95$1 * w), $MachinePrecision] * N[(N[Abs[r], $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.8949999999999997e-43

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 85.2%

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

   3. Applied rewrites 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 92.9%

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

   8. Applied rewrites 92.9%

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

   if 3.8949999999999997e-43 < r

   1. Initial program 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 87.7%

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

   3. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 96.6%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.7%

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 99.7%

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

7. Applied rewrites 99.7%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (fma(-2.0, v, 3.0) * (0.125 * ((w * r) * ((w * 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(fma(-2.0, v, 3.0) * Float64(0.125 * Float64(Float64(w * r) * Float64(Float64(w * r) / Float64(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[(-2.0 * v + 3.0), $MachinePrecision] * N[(0.125 * N[(N[(w * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.7%

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

5. Applied rewrites 99.7%

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

Alternative 5: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 85.2%

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

3. Applied rewrites 90.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. div-flip-rev N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. lift-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

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

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

   16. sub-negate-rev N/A

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

   17. lift-/.f64 N/A

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

   18. frac-2neg N/A

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

   19. distribute-frac-neg2 N/A

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

   20. distribute-neg-frac N/A

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

   21. lift-/.f64 N/A

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

   22. frac-2neg-rev N/A

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

   23. lower-/.f64 N/A

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

7. Applied rewrites 99.7%

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

Alternative 6: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(w \cdot 0.125\right) \cdot \left|r\right|\\ \mathbf{if}\;\left|r\right| \leq 1700:\\ \;\;\;\;-\mathsf{fma}\left(t\_0, 2 \cdot \left(\left|r\right| \cdot w\right), 1.5 - \frac{2}{\left|r\right| \cdot \left|r\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(t\_0, \left(\frac{\left|r\right|}{1 - v} \cdot w\right) \cdot \mathsf{fma}\left(v, -2, 3\right), 1.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w 0.125) (fabs r))))
   (if (<= (fabs r) 1700.0)
     (- (fma t_0 (* 2.0 (* (fabs r) w)) (- 1.5 (/ 2.0 (* (fabs r) (fabs r))))))
     (- (fma t_0 (* (* (/ (fabs r) (- 1.0 v)) w) (fma v -2.0 3.0)) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (w * 0.125) * fabs(r);
	double tmp;
	if (fabs(r) <= 1700.0) {
		tmp = -fma(t_0, (2.0 * (fabs(r) * w)), (1.5 - (2.0 / (fabs(r) * fabs(r)))));
	} else {
		tmp = -fma(t_0, (((fabs(r) / (1.0 - v)) * w) * fma(v, -2.0, 3.0)), 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * 0.125) * abs(r))
	tmp = 0.0
	if (abs(r) <= 1700.0)
		tmp = Float64(-fma(t_0, Float64(2.0 * Float64(abs(r) * w)), Float64(1.5 - Float64(2.0 / Float64(abs(r) * abs(r))))));
	else
		tmp = Float64(-fma(t_0, Float64(Float64(Float64(abs(r) / Float64(1.0 - v)) * w) * fma(v, -2.0, 3.0)), 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * 0.125), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 1700.0], (-N[(t$95$0 * N[(2.0 * N[(N[Abs[r], $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] + N[(1.5 - N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), (-N[(t$95$0 * N[(N[(N[(N[Abs[r], $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if r < 1700

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 85.2%

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

   3. Applied rewrites 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 92.9%

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

   8. Applied rewrites 92.9%

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

   if 1700 < r

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 85.2%

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

   3. Applied rewrites 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 54.8%

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

Alternative 7: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 85.2%

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

3. Applied rewrites 90.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

5. Applied rewrites 99.3%

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

Alternative 8: 92.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 85.2%

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

3. Applied rewrites 90.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 92.9%

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

8. Applied rewrites 92.9%

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

Alternative 9: 58.0% accurate, 2.4× speedup

Math

\[-\left(1 - \frac{\frac{2}{r \cdot r}}{1.5}\right) \cdot 1.5 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -((1.0d0 - ((2.0d0 / (r * r)) / 1.5d0)) * 1.5d0)
end function

Java

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

Python

def code(v, w, r):
	return -((1.0 - ((2.0 / (r * r)) / 1.5)) * 1.5)

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = -((1.0 - ((2.0 / (r * r)) / 1.5)) * 1.5);
end

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 85.2%

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

3. Applied rewrites 90.4%

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

4. Taylor expanded in w around 0

   \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 58.0%

      \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

6. Applied rewrites 58.0%

   \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

8. Applied rewrites 58.0%

   \[\leadsto -\color{blue}{\left(1 - \frac{\frac{2}{r \cdot r}}{1.5}\right) \cdot 1.5} \]
9. Add Preprocessing

Alternative 10: 58.0% accurate, 4.2× speedup

Math

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

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 85.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 85.2%

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

3. Applied rewrites 90.4%

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

4. Taylor expanded in w around 0

   \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 58.0%

      \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

6. Applied rewrites 58.0%

   \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{3}{2} - \frac{2}{{r}^{2}}\right)\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} - \frac{2}{{r}^{2}}\right)}\right) \]

   3. sub-negate N/A

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

   4. lower--.f64 58.0%

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 58.0%

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

8. Applied rewrites 58.0%

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

Alternative 11: 45.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 45.0%

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

4. Applied rewrites 45.0%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 45.0%

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

6. Applied rewrites 45.0%

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

Reproduce

herbie shell --seed 2025191 
(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))