Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.8%

Time: 6.0s

Alternatives: 13

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 50000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(v, 2, -3\right), \frac{\left(\left(0.125 \cdot w\right) \cdot \left(r\_m \cdot w\right)\right) \cdot r\_m}{1 - v}, -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 50000000.0)
   (-
    (-
     (+ 3.0 (/ 2.0 (* r_m r_m)))
     (* w (* (* (* r_m w) r_m) (/ (fma v -0.25 0.375) (- 1.0 v)))))
    4.5)
   (fma
    (fma v 2.0 -3.0)
    (/ (* (* (* 0.125 w) (* r_m w)) r_m) (- 1.0 v))
    -1.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 50000000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (w * (((r_m * w) * r_m) * (fma(v, -0.25, 0.375) / (1.0 - v))))) - 4.5;
	} else {
		tmp = fma(fma(v, 2.0, -3.0), ((((0.125 * w) * (r_m * w)) * r_m) / (1.0 - v)), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 50000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(w * Float64(Float64(Float64(r_m * w) * r_m) * Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v))))) - 4.5);
	else
		tmp = fma(fma(v, 2.0, -3.0), Float64(Float64(Float64(Float64(0.125 * w) * Float64(r_m * w)) * r_m) / Float64(1.0 - v)), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 50000000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(N[(N[(N[(0.125 * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5e7

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   if 5e7 < r

   1. Initial program 85.3%

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in r around inf

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

   6. Applied rewrites 55.2%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r_m r_m)))
   (* (* r_m w) (* (* r_m w) (/ (fma v -0.25 0.375) (- 1.0 v)))))
  4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * ((r_m * w) * (fma(v, -0.25, 0.375) / (1.0 - v))))) - 4.5;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v))))) - 4.5)
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Applied rewrites 99.8%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := 3 + \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(t\_0 - \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot 0.25\right)\right) - 4.5\\ \mathbf{if}\;v \leq -3.25 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\left(t\_0 - \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r_m r_m))))
        (t_1 (- (- t_0 (* (* r_m w) (* (* r_m w) 0.25))) 4.5)))
   (if (<= v -3.25e+41)
     t_1
     (if (<= v 3.9e-19)
       (- (- t_0 (* (* r_m w) (* (* r_m w) 0.375))) 4.5)
       t_1))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double t_1 = (t_0 - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_1;
	} else if (v <= 3.9e-19) {
		tmp = (t_0 - ((r_m * w) * ((r_m * w) * 0.375))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r_m * r_m))
    t_1 = (t_0 - ((r_m * w) * ((r_m * w) * 0.25d0))) - 4.5d0
    if (v <= (-3.25d+41)) then
        tmp = t_1
    else if (v <= 3.9d-19) then
        tmp = (t_0 - ((r_m * w) * ((r_m * w) * 0.375d0))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double t_1 = (t_0 - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_1;
	} else if (v <= 3.9e-19) {
		tmp = (t_0 - ((r_m * w) * ((r_m * w) * 0.375))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 3.0 + (2.0 / (r_m * r_m))
	t_1 = (t_0 - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5
	tmp = 0
	if v <= -3.25e+41:
		tmp = t_1
	elif v <= 3.9e-19:
		tmp = (t_0 - ((r_m * w) * ((r_m * w) * 0.375))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	t_1 = Float64(Float64(t_0 - Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * 0.25))) - 4.5)
	tmp = 0.0
	if (v <= -3.25e+41)
		tmp = t_1;
	elseif (v <= 3.9e-19)
		tmp = Float64(Float64(t_0 - Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * 0.375))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 3.0 + (2.0 / (r_m * r_m));
	t_1 = (t_0 - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	tmp = 0.0;
	if (v <= -3.25e+41)
		tmp = t_1;
	elseif (v <= 3.9e-19)
		tmp = (t_0 - ((r_m * w) * ((r_m * w) * 0.375))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -3.25e+41], t$95$1, If[LessEqual[v, 3.9e-19], N[(N[(t$95$0 - N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.24999999999999988e41 or 3.89999999999999995e-19 < v

   1. Initial program 85.3%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 93.3%

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

   if -3.24999999999999988e41 < v < 3.89999999999999995e-19

   1. Initial program 85.3%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 93.3%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4000000:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(v, 2, -3\right), \frac{\left(\left(0.125 \cdot w\right) \cdot \left(r\_m \cdot w\right)\right) \cdot r\_m}{1 - v}, -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4000000.0)
   (- (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) (* w (* (* (* r_m w) r_m) 0.375))) 4.5)
   (fma
    (fma v 2.0 -3.0)
    (/ (* (* (* 0.125 w) (* r_m w)) r_m) (- 1.0 v))
    -1.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4000000.0) {
		tmp = ((3.0 + ((2.0 / r_m) / r_m)) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	} else {
		tmp = fma(fma(v, 2.0, -3.0), ((((0.125 * w) * (r_m * w)) * r_m) / (1.0 - v)), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - Float64(w * Float64(Float64(Float64(r_m * w) * r_m) * 0.375))) - 4.5);
	else
		tmp = fma(fma(v, 2.0, -3.0), Float64(Float64(Float64(Float64(0.125 * w) * Float64(r_m * w)) * r_m) / Float64(1.0 - v)), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4000000.0], N[(N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(N[(N[(N[(0.125 * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4e6

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

   if 4e6 < r

   1. Initial program 85.3%

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in r around inf

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

   6. Applied rewrites 55.2%

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

Alternative 5: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4000000:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(v, 2, -3\right), \left(r\_m \cdot w\right) \cdot \left(\left(0.125 \cdot w\right) \cdot \frac{r\_m}{1 - v}\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4000000.0)
   (- (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) (* w (* (* (* r_m w) r_m) 0.375))) 4.5)
   (fma
    (fma v 2.0 -3.0)
    (* (* r_m w) (* (* 0.125 w) (/ r_m (- 1.0 v))))
    -1.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4000000.0) {
		tmp = ((3.0 + ((2.0 / r_m) / r_m)) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	} else {
		tmp = fma(fma(v, 2.0, -3.0), ((r_m * w) * ((0.125 * w) * (r_m / (1.0 - v)))), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - Float64(w * Float64(Float64(Float64(r_m * w) * r_m) * 0.375))) - 4.5);
	else
		tmp = fma(fma(v, 2.0, -3.0), Float64(Float64(r_m * w) * Float64(Float64(0.125 * w) * Float64(r_m / Float64(1.0 - v)))), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4000000.0], N[(N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(v * 2.0 + -3.0), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(0.125 * w), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4e6

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

   if 4e6 < r

   1. Initial program 85.3%

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in r around inf

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

   6. Applied rewrites 55.2%

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

   8. Applied rewrites 55.8%

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

Alternative 6: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(\left(3 + t\_0\right) - \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot 0.25\right)\right) - 4.5\\ \mathbf{if}\;v \leq -3.25 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\left(\left(t\_0 - -3\right) - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1 (- (- (+ 3.0 t_0) (* (* r_m w) (* (* r_m w) 0.25))) 4.5)))
   (if (<= v -3.25e+41)
     t_1
     (if (<= v 3.9e-19)
       (- (- (- t_0 -3.0) (* w (* (* (* r_m w) r_m) 0.375))) 4.5)
       t_1))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = ((3.0 + t_0) - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_1;
	} else if (v <= 3.9e-19) {
		tmp = ((t_0 - -3.0) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    t_1 = ((3.0d0 + t_0) - ((r_m * w) * ((r_m * w) * 0.25d0))) - 4.5d0
    if (v <= (-3.25d+41)) then
        tmp = t_1
    else if (v <= 3.9d-19) then
        tmp = ((t_0 - (-3.0d0)) - (w * (((r_m * w) * r_m) * 0.375d0))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = ((3.0 + t_0) - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_1;
	} else if (v <= 3.9e-19) {
		tmp = ((t_0 - -3.0) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = ((3.0 + t_0) - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5
	tmp = 0
	if v <= -3.25e+41:
		tmp = t_1
	elif v <= 3.9e-19:
		tmp = ((t_0 - -3.0) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * 0.25))) - 4.5)
	tmp = 0.0
	if (v <= -3.25e+41)
		tmp = t_1;
	elseif (v <= 3.9e-19)
		tmp = Float64(Float64(Float64(t_0 - -3.0) - Float64(w * Float64(Float64(Float64(r_m * w) * r_m) * 0.375))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	t_1 = ((3.0 + t_0) - ((r_m * w) * ((r_m * w) * 0.25))) - 4.5;
	tmp = 0.0;
	if (v <= -3.25e+41)
		tmp = t_1;
	elseif (v <= 3.9e-19)
		tmp = ((t_0 - -3.0) - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -3.25e+41], t$95$1, If[LessEqual[v, 3.9e-19], N[(N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.24999999999999988e41 or 3.89999999999999995e-19 < v

   1. Initial program 85.3%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 93.3%

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

   if -3.24999999999999988e41 < v < 3.89999999999999995e-19

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

Alternative 7: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \left(r\_m \cdot w\right) \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m} - -3\\ t_2 := \left(t\_1 - w \cdot \left(t\_0 \cdot 0.25\right)\right) - 4.5\\ \mathbf{if}\;v \leq -3.25 \cdot 10^{+41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\left(t\_1 - w \cdot \left(t\_0 \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* r_m w) r_m))
        (t_1 (- (/ 2.0 (* r_m r_m)) -3.0))
        (t_2 (- (- t_1 (* w (* t_0 0.25))) 4.5)))
   (if (<= v -3.25e+41)
     t_2
     (if (<= v 3.9e-19) (- (- t_1 (* w (* t_0 0.375))) 4.5) t_2))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * r_m;
	double t_1 = (2.0 / (r_m * r_m)) - -3.0;
	double t_2 = (t_1 - (w * (t_0 * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_2;
	} else if (v <= 3.9e-19) {
		tmp = (t_1 - (w * (t_0 * 0.375))) - 4.5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (r_m * w) * r_m
    t_1 = (2.0d0 / (r_m * r_m)) - (-3.0d0)
    t_2 = (t_1 - (w * (t_0 * 0.25d0))) - 4.5d0
    if (v <= (-3.25d+41)) then
        tmp = t_2
    else if (v <= 3.9d-19) then
        tmp = (t_1 - (w * (t_0 * 0.375d0))) - 4.5d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * r_m;
	double t_1 = (2.0 / (r_m * r_m)) - -3.0;
	double t_2 = (t_1 - (w * (t_0 * 0.25))) - 4.5;
	double tmp;
	if (v <= -3.25e+41) {
		tmp = t_2;
	} else if (v <= 3.9e-19) {
		tmp = (t_1 - (w * (t_0 * 0.375))) - 4.5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (r_m * w) * r_m
	t_1 = (2.0 / (r_m * r_m)) - -3.0
	t_2 = (t_1 - (w * (t_0 * 0.25))) - 4.5
	tmp = 0
	if v <= -3.25e+41:
		tmp = t_2
	elif v <= 3.9e-19:
		tmp = (t_1 - (w * (t_0 * 0.375))) - 4.5
	else:
		tmp = t_2
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(r_m * w) * r_m)
	t_1 = Float64(Float64(2.0 / Float64(r_m * r_m)) - -3.0)
	t_2 = Float64(Float64(t_1 - Float64(w * Float64(t_0 * 0.25))) - 4.5)
	tmp = 0.0
	if (v <= -3.25e+41)
		tmp = t_2;
	elseif (v <= 3.9e-19)
		tmp = Float64(Float64(t_1 - Float64(w * Float64(t_0 * 0.375))) - 4.5);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (r_m * w) * r_m;
	t_1 = (2.0 / (r_m * r_m)) - -3.0;
	t_2 = (t_1 - (w * (t_0 * 0.25))) - 4.5;
	tmp = 0.0;
	if (v <= -3.25e+41)
		tmp = t_2;
	elseif (v <= 3.9e-19)
		tmp = (t_1 - (w * (t_0 * 0.375))) - 4.5;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(w * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -3.25e+41], t$95$2, If[LessEqual[v, 3.9e-19], N[(N[(t$95$1 - N[(w * N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.24999999999999988e41 or 3.89999999999999995e-19 < v

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.8%

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

   if -3.24999999999999988e41 < v < 3.89999999999999995e-19

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

Alternative 8: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \left(r\_m \cdot w\right) \cdot r\_m\\ \mathbf{if}\;r\_m \leq 2.55 \cdot 10^{+38}:\\ \;\;\;\;\left(\left(\frac{2}{r\_m \cdot r\_m} - -3\right) - w \cdot \left(t\_0 \cdot 0.25\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - w \cdot \left(t\_0 \cdot 0.375\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* r_m w) r_m)))
   (if (<= r_m 2.55e+38)
     (- (- (- (/ 2.0 (* r_m r_m)) -3.0) (* w (* t_0 0.25))) 4.5)
     (- (- 3.0 (* w (* t_0 0.375))) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * r_m;
	double tmp;
	if (r_m <= 2.55e+38) {
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (t_0 * 0.25))) - 4.5;
	} else {
		tmp = (3.0 - (w * (t_0 * 0.375))) - 4.5;
	}
	return tmp;
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r_m * w) * r_m
    if (r_m <= 2.55d+38) then
        tmp = (((2.0d0 / (r_m * r_m)) - (-3.0d0)) - (w * (t_0 * 0.25d0))) - 4.5d0
    else
        tmp = (3.0d0 - (w * (t_0 * 0.375d0))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = (r_m * w) * r_m;
	double tmp;
	if (r_m <= 2.55e+38) {
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (t_0 * 0.25))) - 4.5;
	} else {
		tmp = (3.0 - (w * (t_0 * 0.375))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (r_m * w) * r_m
	tmp = 0
	if r_m <= 2.55e+38:
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (t_0 * 0.25))) - 4.5
	else:
		tmp = (3.0 - (w * (t_0 * 0.375))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(r_m * w) * r_m)
	tmp = 0.0
	if (r_m <= 2.55e+38)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r_m * r_m)) - -3.0) - Float64(w * Float64(t_0 * 0.25))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(w * Float64(t_0 * 0.375))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (r_m * w) * r_m;
	tmp = 0.0;
	if (r_m <= 2.55e+38)
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (t_0 * 0.25))) - 4.5;
	else
		tmp = (3.0 - (w * (t_0 * 0.375))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision]}, If[LessEqual[r$95$m, 2.55e+38], N[(N[(N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - N[(w * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(w * N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.5500000000000001e38

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.8%

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

   if 2.5500000000000001e38 < r

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 48.3%

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

Alternative 9: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\frac{2}{r\_m \cdot r\_m} - -3\right) - w \cdot \left(\left(\left(r\_m \cdot r\_m\right) \cdot w\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot 0.375\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 5e+138)
   (- (- (- (/ 2.0 (* r_m r_m)) -3.0) (* w (* (* (* r_m r_m) w) 0.375))) 4.5)
   (- (- 3.0 (* w (* (* (* r_m w) r_m) 0.375))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5e+138) {
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (((r_m * r_m) * w) * 0.375))) - 4.5;
	} else {
		tmp = (3.0 - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	}
	return tmp;
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 5d+138) then
        tmp = (((2.0d0 / (r_m * r_m)) - (-3.0d0)) - (w * (((r_m * r_m) * w) * 0.375d0))) - 4.5d0
    else
        tmp = (3.0d0 - (w * (((r_m * w) * r_m) * 0.375d0))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5e+138) {
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (((r_m * r_m) * w) * 0.375))) - 4.5;
	} else {
		tmp = (3.0 - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 5e+138:
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (((r_m * r_m) * w) * 0.375))) - 4.5
	else:
		tmp = (3.0 - (w * (((r_m * w) * r_m) * 0.375))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 5e+138)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r_m * r_m)) - -3.0) - Float64(w * Float64(Float64(Float64(r_m * r_m) * w) * 0.375))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(w * Float64(Float64(Float64(r_m * w) * r_m) * 0.375))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 5e+138)
		tmp = (((2.0 / (r_m * r_m)) - -3.0) - (w * (((r_m * r_m) * w) * 0.375))) - 4.5;
	else
		tmp = (3.0 - (w * (((r_m * w) * r_m) * 0.375))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 5e+138], N[(N[(N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - N[(w * N[(N[(N[(r$95$m * r$95$m), $MachinePrecision] * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.00000000000000016e138

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   8. Applied rewrites 91.8%

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

   10. Applied rewrites 86.7%

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

   if 5.00000000000000016e138 < r

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 48.3%

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

Alternative 10: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - w \cdot \left(\left(\left(r\_m \cdot w\right) \cdot r\_m\right) \cdot 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r_m) r_m))
           (- 1.0 v)))
         4.5)
        -1.5)
     (- (- 3.0 (* w (* (* (* r_m w) r_m) 0.375))) 4.5)
     (- t_0 1.5))))

C

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

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r_m) * r_m)) / (1.0d0 - v))) - 4.5d0) <= (-1.5d0)) then
        tmp = (3.0d0 - (w * (((r_m * w) * r_m) * 0.375d0))) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(w * N[(N[(N[(r$95$m * w), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.3%

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

   3. Applied rewrites 97.1%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 48.3%

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

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

   1. Initial program 85.3%

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

   3. Applied rewrites 81.3%

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

   4. Taylor expanded in w around 0

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

   6. Applied rewrites 57.2%

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

   8. Applied rewrites 57.2%

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

Alternative 11: 57.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} - 1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- (/ 2.0 (* r_m r_m)) 1.5))

C

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

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) - 1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 / (r_m * r_m)) - 1.5

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Applied rewrites 81.3%

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

4. Taylor expanded in w around 0

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

6. Applied rewrites 57.2%

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

8. Applied rewrites 57.2%

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

Alternative 12: 44.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{\frac{2}{r\_m}}{r\_m} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (/ (/ 2.0 r_m) r_m))

C

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

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (2.0d0 / r_m) / r_m
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 / r_m) / r_m;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 / r_m) / r_m

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in r around 0

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

4. Applied rewrites 44.1%

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

6. Applied rewrites 44.1%

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

8. Applied rewrites 44.1%

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

Alternative 13: 44.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (/ 2.0 (* r_m r_m)))

C

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

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return 2.0 / (r_m * r_m);
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return 2.0 / (r_m * r_m)

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in r around 0

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

4. Applied rewrites 44.1%

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

6. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2025161 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))