Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.7%

Time: 4.3s

Alternatives: 18

Speedup: 1.8×

• 52.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Applied rewrites 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_1 := \frac{2}{r \cdot r} + 3\\ t_2 := t\_1 - \mathsf{fma}\left(-0.25 \cdot v, t\_0, 4.5\right)\\ \mathbf{if}\;v \leq -7800000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 0.045:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(0.375, t\_0, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (* (* r w) (* r w)) (- 1.0 v)))
        (t_1 (+ (/ 2.0 (* r r)) 3.0))
        (t_2 (- t_1 (fma (* -0.25 v) t_0 4.5))))
   (if (<= v -7800000000.0)
     t_2
     (if (<= v 0.045) (- t_1 (fma 0.375 t_0 4.5)) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) / (1.0 - v);
	double t_1 = (2.0 / (r * r)) + 3.0;
	double t_2 = t_1 - fma((-0.25 * v), t_0, 4.5);
	double tmp;
	if (v <= -7800000000.0) {
		tmp = t_2;
	} else if (v <= 0.045) {
		tmp = t_1 - fma(0.375, t_0, 4.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	t_2 = Float64(t_1 - fma(Float64(-0.25 * v), t_0, 4.5))
	tmp = 0.0
	if (v <= -7800000000.0)
		tmp = t_2;
	elseif (v <= 0.045)
		tmp = Float64(t_1 - fma(0.375, t_0, 4.5));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(-0.25 * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -7800000000.0], t$95$2, If[LessEqual[v, 0.045], N[(t$95$1 - N[(0.375 * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.8e9 or 0.044999999999999998 < v

   1. Initial program 81.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -7.8e9 < v < 0.044999999999999998

   1. Initial program 88.0%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 99.2%

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

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -50000000000000:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(\left(\frac{0.375}{v} - 0.25\right) \cdot v, t\_0, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + 3\right) - \mathsf{fma}\left(-0.25 \cdot v, t\_0, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (* (* r w) (* r w)) (- 1.0 v))) (t_1 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_1)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -50000000000000.0)
     (- t_1 (fma (* (- (/ 0.375 v) 0.25) v) t_0 4.5))
     (- (+ t_1 3.0) (fma (* -0.25 v) t_0 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) / (1.0 - v);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -50000000000000.0) {
		tmp = t_1 - fma((((0.375 / v) - 0.25) * v), t_0, 4.5);
	} else {
		tmp = (t_1 + 3.0) - fma((-0.25 * v), t_0, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -50000000000000.0)
		tmp = Float64(t_1 - fma(Float64(Float64(Float64(0.375 / v) - 0.25) * v), t_0, 4.5));
	else
		tmp = Float64(Float64(t_1 + 3.0) - fma(Float64(-0.25 * v), t_0, 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -50000000000000.0], N[(t$95$1 - N[(N[(N[(N[(0.375 / v), $MachinePrecision] - 0.25), $MachinePrecision] * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + 3.0), $MachinePrecision] - N[(N[(-0.25 * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $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)) < -5e13

   1. Initial program 85.7%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \mathsf{fma}\left(\left(\frac{\frac{3}{8}}{v} - \frac{1}{4}\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -5e13 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.4%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

Alternative 4: 98.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

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

   1. Initial program 87.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

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

   1. Initial program 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 w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \mathbf{if}\;v \leq -7800000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 0.045:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))
   (if (<= v -7800000000.0)
     t_1
     (if (<= v 0.045)
       (- (+ t_0 3.0) (fma 0.375 (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double tmp;
	if (v <= -7800000000.0) {
		tmp = t_1;
	} else if (v <= 0.045) {
		tmp = (t_0 + 3.0) - fma(0.375, (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	tmp = 0.0
	if (v <= -7800000000.0)
		tmp = t_1;
	elseif (v <= 0.045)
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -7800000000.0], t$95$1, If[LessEqual[v, 0.045], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.8e9 or 0.044999999999999998 < v

   1. Initial program 81.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if -7.8e9 < v < 0.044999999999999998

   1. Initial program 88.0%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 99.2%

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

Alternative 6: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ t_2 := \left(w \cdot w\right) \cdot r\\ t_3 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_2 \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -50000000000000:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_2\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5)))
        (t_2 (* (* w w) r))
        (t_3
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_2 r)) (- 1.0 v)))
          4.5)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -50000000000000.0)
       (- (- 3.0 (/ (* (* (* (fma v -2.0 3.0) 0.125) t_2) r) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double t_2 = (w * w) * r;
	double t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (t_2 * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -50000000000000.0) {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * t_2) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	t_2 = Float64(Float64(w * w) * r)
	t_3 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_2 * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -50000000000000.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_2) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -50000000000000.0], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$2), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0 or -5e13 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 83.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -5e13

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 97.3%

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

Alternative 7: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := t\_0 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -50000000000000:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5))
        (t_2 (- t_0 (fma (* (* (* w r) r) w) 0.25 1.5))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -50000000000000.0)
       (* (/ (* (* (* (fma v -2.0 3.0) (* w w)) r) r) (- 1.0 v)) -0.125)
       t_2))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double t_2 = t_0 - fma((((w * r) * r) * w), 0.25, 1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -50000000000000.0) {
		tmp = ((((fma(v, -2.0, 3.0) * (w * w)) * r) * r) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	t_2 = Float64(t_0 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -50000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * Float64(w * w)) * r) * r) / Float64(1.0 - v)) * -0.125);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -50000000000000.0], N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0 or -5e13 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 83.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -5e13

   1. Initial program 98.6%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in w around inf

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

   9. Applied rewrites 70.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. *-commutative N/A

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

       14. +-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lift-fma.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 96.6

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

   11. Applied rewrites 96.6%

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

Alternative 8: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(w \cdot r\right) \cdot r\right) \cdot w\\ t_2 := t\_0 - \mathsf{fma}\left(t\_1, 0.25, 1.5\right)\\ \mathbf{if}\;v \leq -7800000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 0.045:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, t\_1, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* (* (* w r) r) w))
        (t_2 (- t_0 (fma t_1 0.25 1.5))))
   (if (<= v -7800000000.0)
     t_2
     (if (<= v 0.045) (- (+ t_0 3.0) (fma 0.375 t_1 4.5)) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((w * r) * r) * w;
	double t_2 = t_0 - fma(t_1, 0.25, 1.5);
	double tmp;
	if (v <= -7800000000.0) {
		tmp = t_2;
	} else if (v <= 0.045) {
		tmp = (t_0 + 3.0) - fma(0.375, t_1, 4.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(w * r) * r) * w)
	t_2 = Float64(t_0 - fma(t_1, 0.25, 1.5))
	tmp = 0.0
	if (v <= -7800000000.0)
		tmp = t_2;
	elseif (v <= 0.045)
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, t_1, 4.5));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(t$95$1 * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -7800000000.0], t$95$2, If[LessEqual[v, 0.045], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * t$95$1 + 4.5), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.8e9 or 0.044999999999999998 < v

   1. Initial program 81.7%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if -7.8e9 < v < 0.044999999999999998

   1. Initial program 88.0%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in v around 0

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

      1. pow-prod-down N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.3

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

   9. Applied rewrites 96.3%

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

Alternative 9: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_0}{1 - v}\right) - 4.5\\ t_3 := t\_1 - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w, 0.25, 1.5\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+174}:\\ \;\;\;\;\left(3 - \frac{\left(0.125 \cdot 3\right) \cdot t\_0}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (-
          (- (+ 3.0 t_1) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_0) (- 1.0 v)))
          4.5))
        (t_3 (- t_1 (fma (* (* (* w r) r) w) 0.25 1.5))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -4e+174) (- (- 3.0 (/ (* (* 0.125 3.0) t_0) 1.0)) 4.5) t_3))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5;
	double t_3 = t_1 - fma((((w * r) * r) * w), 0.25, 1.5);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -4e+174) {
		tmp = (3.0 - (((0.125 * 3.0) * t_0) / 1.0)) - 4.5;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_0) / Float64(1.0 - v))) - 4.5)
	t_3 = Float64(t_1 - fma(Float64(Float64(Float64(w * r) * r) * w), 0.25, 1.5))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -4e+174)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.125 * 3.0) * t_0) / 1.0)) - 4.5);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25 + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -4e+174], N[(N[(3.0 - N[(N[(N[(0.125 * 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0 or -4.00000000000000028e174 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.5%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -4.00000000000000028e174

   1. Initial program 99.0%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 84.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 86.5%

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

Alternative 10: 91.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 8 \cdot 10^{-71}:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;r \leq 1.28 \cdot 10^{+39}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.375 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.25\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 8e-71)
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= r 1.28e+39)
       (- t_0 (fma (* 0.375 (* r r)) (* w w) 1.5))
       (- (- 3.0 (* (* (* (* w r) r) w) 0.25)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8e-71) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (r <= 1.28e+39) {
		tmp = t_0 - fma((0.375 * (r * r)), (w * w), 1.5);
	} else {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 8e-71)
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (r <= 1.28e+39)
		tmp = Float64(t_0 - fma(Float64(0.375 * Float64(r * r)), Float64(w * w), 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * r) * w) * 0.25)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 8e-71], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.28e+39], N[(t$95$0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.9999999999999993e-71

   1. Initial program 83.2%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 83.7

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

   7. Applied rewrites 83.7%

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

   if 7.9999999999999993e-71 < r < 1.27999999999999994e39

   1. Initial program 91.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 1.27999999999999994e39 < r

   1. Initial program 87.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in v around inf

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 73.4%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w}\right) - 4.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 84.0%

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

Alternative 11: 87.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;r \leq 1.28 \cdot 10^{+39}:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.25\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.1e-30)
   (- (/ 2.0 (* r r)) (* (* (* (* r r) 0.25) w) w))
   (if (<= r 1.28e+39)
     (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
     (- (- 3.0 (* (* (* (* w r) r) w) 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.1e-30) {
		tmp = (2.0 / (r * r)) - ((((r * r) * 0.25) * w) * w);
	} else if (r <= 1.28e+39) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.1d-30) then
        tmp = (2.0d0 / (r * r)) - ((((r * r) * 0.25d0) * w) * w)
    else if (r <= 1.28d+39) then
        tmp = (3.0d0 - ((0.375d0 * (r * r)) * (w * w))) - 4.5d0
    else
        tmp = (3.0d0 - ((((w * r) * r) * w) * 0.25d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.1e-30) {
		tmp = (2.0 / (r * r)) - ((((r * r) * 0.25) * w) * w);
	} else if (r <= 1.28e+39) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.1e-30:
		tmp = (2.0 / (r * r)) - ((((r * r) * 0.25) * w) * w)
	elif r <= 1.28e+39:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.1e-30)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (r <= 1.28e+39)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * r) * w) * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.1e-30)
		tmp = (2.0 / (r * r)) - ((((r * r) * 0.25) * w) * w);
	elseif (r <= 1.28e+39)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.1e-30], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.28e+39], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.09999999999999992e-30

   1. Initial program 83.4%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if 1.09999999999999992e-30 < r < 1.27999999999999994e39

   1. Initial program 92.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in v around 0

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 62.3%

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

   if 1.27999999999999994e39 < r

   1. Initial program 87.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in v around inf

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 73.4%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w}\right) - 4.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 84.0%

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

Alternative 12: 87.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in v around inf

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 78.0

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

4. Applied rewrites 78.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 91.6

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

6. Applied rewrites 91.6%

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

Alternative 13: 86.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.28 \cdot 10^{+39}:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot 0.25\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7.8e-31)
   (/ (/ 2.0 r) r)
   (if (<= r 1.28e+39)
     (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
     (- (- 3.0 (* (* (* (* w r) r) w) 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.8e-31) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1.28e+39) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7.8d-31) then
        tmp = (2.0d0 / r) / r
    else if (r <= 1.28d+39) then
        tmp = (3.0d0 - ((0.375d0 * (r * r)) * (w * w))) - 4.5d0
    else
        tmp = (3.0d0 - ((((w * r) * r) * w) * 0.25d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.8e-31) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1.28e+39) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7.8e-31:
		tmp = (2.0 / r) / r
	elif r <= 1.28e+39:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7.8e-31)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 1.28e+39)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * r) * w) * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7.8e-31)
		tmp = (2.0 / r) / r;
	elseif (r <= 1.28e+39)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = (3.0 - ((((w * r) * r) * w) * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7.8e-31], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 1.28e+39], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.8000000000000003e-31

   1. Initial program 83.4%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 57.9

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

   4. Applied rewrites 57.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.9

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

   6. Applied rewrites 57.9%

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

   if 7.8000000000000003e-31 < r < 1.27999999999999994e39

   1. Initial program 92.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in v around 0

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 62.3%

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

   if 1.27999999999999994e39 < r

   1. Initial program 87.7%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in v around inf

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 73.4%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w}\right) - 4.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 84.0%

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

Alternative 14: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq -500000000000:\\ \;\;\;\;\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 -5e+168)
     (* (* -0.375 (* r r)) (* w w))
     (if (<= t_1 -500000000000.0) (* (* -0.25 (* r r)) (* w w)) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -5e+168) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} else if (t_1 <= -500000000000.0) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
    if (t_1 <= (-5d+168)) then
        tmp = ((-0.375d0) * (r * r)) * (w * w)
    else if (t_1 <= (-500000000000.0d0)) then
        tmp = ((-0.25d0) * (r * r)) * (w * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -5e+168) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} else if (t_1 <= -500000000000.0) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -5e+168:
		tmp = (-0.375 * (r * r)) * (w * w)
	elif t_1 <= -500000000000.0:
		tmp = (-0.25 * (r * r)) * (w * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_1 <= -5e+168)
		tmp = Float64(Float64(-0.375 * Float64(r * r)) * Float64(w * w));
	elseif (t_1 <= -500000000000.0)
		tmp = Float64(Float64(-0.25 * Float64(r * r)) * Float64(w * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -5e+168)
		tmp = (-0.375 * (r * r)) * (w * w);
	elseif (t_1 <= -500000000000.0)
		tmp = (-0.25 * (r * r)) * (w * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+168], N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000000000.0], N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -4.99999999999999967e168

   1. Initial program 84.0%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in w around inf

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

   9. Applied rewrites 80.6%

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

   10. Taylor expanded in v around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 80.8

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

   12. Applied rewrites 80.8%

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

   if -4.99999999999999967e168 < (-.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)) < -5e11

   1. Initial program 98.3%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 40.2

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 39.7

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

   7. Applied rewrites 39.7%

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

   if -5e11 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.4%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

Alternative 15: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5000000000024976:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5000000000024976)
     (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5000000000024976) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1.5000000000024976d0)) then
        tmp = (3.0d0 - ((0.375d0 * (r * r)) * (w * w))) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5000000000024976) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5000000000024976:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.5000000000024976)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5000000000024976)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5000000000024976], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $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)) < -1.5000000000024976

   1. Initial program 85.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in v around 0

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

   7. Applied rewrites 77.8%

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

   if -1.5000000000024976 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.3%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

Alternative 16: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -500000000000:\\ \;\;\;\;\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -500000000000.0)
     (* (* -0.375 (* r r)) (* w w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -500000000000.0) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-500000000000.0d0)) then
        tmp = ((-0.375d0) * (r * r)) * (w * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -500000000000.0) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -500000000000.0:
		tmp = (-0.375 * (r * r)) * (w * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -500000000000.0)
		tmp = Float64(Float64(-0.375 * Float64(r * r)) * Float64(w * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -500000000000.0)
		tmp = (-0.375 * (r * r)) * (w * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -500000000000.0], N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -5e11

   1. Initial program 85.7%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in w around inf

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

   9. Applied rewrites 80.3%

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

   10. Taylor expanded in v around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 77.8

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

   12. Applied rewrites 77.8%

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

   if -5e11 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.4%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

Alternative 17: 57.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 57.0

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

4. Applied rewrites 57.0%

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

Alternative 18: 43.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in r around 0

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

   1. pow2 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 43.7

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

4. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2025110 
(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))