Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 70.0% → 92.5%

Time: 5.3s

Alternatives: 11

Speedup: 1.3×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.5e-140)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (if (<= y_m 1.4e+149)
      (* (fma (/ (* (+ x z) (- x z)) (* y_m y_m)) 0.5 0.5) y_m)
      (* 0.5 (- y_m (* z (/ z y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-140) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 1.4e+149) {
		tmp = fma((((x + z) * (x - z)) / (y_m * y_m)), 0.5, 0.5) * y_m;
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-140)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	elseif (y_m <= 1.4e+149)
		tmp = Float64(fma(Float64(Float64(Float64(x + z) * Float64(x - z)) / Float64(y_m * y_m)), 0.5, 0.5) * y_m);
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.5e-140], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 1.4e+149], N[(N[(N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.50000000000000009e-140

   1. Initial program 89.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 73.7

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 97.5

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 97.5%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   if 1.50000000000000009e-140 < y < 1.4e149

   1. Initial program 89.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 92.5

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   if 1.4e149 < y

   1. Initial program 12.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 72.8

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 87.4

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 87.4%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 1.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4e-142)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (if (<= y_m 1.3e+127)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* 0.5 (- y_m (* z (/ z y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-142) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 1.3e+127) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4d-142) then
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    else if (y_m <= 1.3d+127) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-142) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 1.3e+127) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 4e-142:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	elif y_m <= 1.3e+127:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-142)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	elseif (y_m <= 1.3e+127)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 4e-142)
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	elseif (y_m <= 1.3e+127)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4e-142], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 1.3e+127], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 4.0000000000000002e-142

   1. Initial program 89.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 73.6

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 97.6

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   if 4.0000000000000002e-142 < y < 1.3000000000000001e127

   1. Initial program 91.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   if 1.3000000000000001e127 < y

   1. Initial program 21.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 72.5

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 72.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 86.4

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 86.4%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.4e+41)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.4e+41) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.4d+41) then
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.4e+41) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.4e+41:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.4e+41)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.4e+41)
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.4e+41], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.39999999999999998e41

   1. Initial program 91.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 84.8

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 88.9

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 88.9%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   if 3.39999999999999998e41 < y

   1. Initial program 41.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 70.6

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 81.4

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 81.4%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \left(z + x\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.4e+41)
    (/ (* (- x z) (+ z x)) (+ y_m y_m))
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.4e+41) {
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.4d+41) then
        tmp = ((x - z) * (z + x)) / (y_m + y_m)
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.4e+41) {
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.4e+41:
		tmp = ((x - z) * (z + x)) / (y_m + y_m)
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.4e+41)
		tmp = Float64(Float64(Float64(x - z) * Float64(z + x)) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.4e+41)
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.4e+41], N[(N[(N[(x - z), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.39999999999999998e41

   1. Initial program 91.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} - {z}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x - {\color{blue}{z}}^{2}}{y \cdot 2} \]

      2. pow2 N/A

         \[\leadsto \frac{x \cdot x - z \cdot \color{blue}{z}}{y \cdot 2} \]

      3. difference-of-squares N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \color{blue}{\left(x - z\right)}}{y \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \color{blue}{\left(x - z\right)}}{y \cdot 2} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(\color{blue}{x} - z\right)}{y \cdot 2} \]

      6. lower--.f64 87.4

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - \color{blue}{z}\right)}{y \cdot 2} \]

   4. Applied rewrites 87.4%

      \[\leadsto \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \color{blue}{\left(x - z\right)}}{y \cdot 2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(\color{blue}{x} - z\right)}{y \cdot 2} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - \color{blue}{z}\right)}{y \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \color{blue}{\left(x + z\right)}}{y \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \color{blue}{\left(x + z\right)}}{y \cdot 2} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \left(\color{blue}{x} + z\right)}{y \cdot 2} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{y \cdot 2} \]

      8. lower-+.f64 87.4

         \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{y \cdot 2} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(y \cdot 2\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite<=}\left(*-commutative, \left(2 \cdot y\right)\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite=>}\left(count-2-rev, \left(y + y\right)\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + y\right)\right)} \]

   6. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{\left(x - z\right) \cdot \left(z + x\right)}{y + y}} \]

   if 3.39999999999999998e41 < y

   1. Initial program 41.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 70.6

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 81.4

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 81.4%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* 0.5 (- y_m (* z (/ z y_m)))))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 INFINITY) (/ (fma y_m y_m (* x x)) (+ y_m y_m)) t_0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = 0.5 * (y_m - (z * (z / y_m)));
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y_m, y_m, (x * x)) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y_m, y_m, Float64(x * x)) / Float64(y_m + y_m));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 0.0 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 68.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 78.1

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 78.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 91.5

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 91.5%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

      2. lift-*.f64 47.2

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

   4. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

      4. lift-+.f64 47.2

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   6. Applied rewrites 47.2%

      \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   7. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} + {y}^{2}}}{y + y} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{y}^{2} + \color{blue}{{x}^{2}}}{y + y} \]

      2. pow2 N/A

         \[\leadsto \frac{y \cdot y + {\color{blue}{x}}^{2}}{y + y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y}, {x}^{2}\right)}{y + y} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} \]

      5. lift-*.f64 70.8

         \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} \]

   9. Applied rewrites 70.8%

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

Alternative 6: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.9e+116)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (* (* x (/ (- x z) y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.9e+116) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.9d+116) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * ((x - z) / y_m)) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.9e+116) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.9e+116:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * ((x - z) / y_m)) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.9e+116)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * Float64(Float64(x - z) / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.9e+116)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * ((x - z) / y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.9e+116], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e116

   1. Initial program 71.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 68.5

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 73.9

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 73.9%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

   if 1.8999999999999999e116 < x

   1. Initial program 60.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 85.6

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 85.6%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 4e+116) (* 0.5 (- y_m (* z (/ z y_m)))) (* (* x (/ x y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 4e+116) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * (x / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4d+116) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * (x / y_m)) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 4e+116) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * (x / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 4e+116:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * (x / y_m)) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 4e+116)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * Float64(x / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 4e+116)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * (x / y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 4e+116], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000006e116

   1. Initial program 71.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2}} - \frac{{z}^{2}}{y \cdot 2} \]

      12. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      13. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} + {x}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      15. pow2 N/A

          \[\leadsto \frac{\color{blue}{y \cdot y} + {x}^{2}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, {x}^{2}\right)}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      17. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right)}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{2 \cdot y}} - \frac{{z}^{2}}{y \cdot 2} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\color{blue}{y + y}} - \frac{{z}^{2}}{y \cdot 2} \]

   3. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{y + y} - z \cdot \frac{z}{y + y}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y - \frac{1}{2} \cdot \frac{{z}^{2}}{y}} \]
   5. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      6. lower-*.f64 68.5

         \[\leadsto 0.5 \cdot \left(y - \frac{z \cdot z}{y}\right) \]

   6. Applied rewrites 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{y}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - \frac{z \cdot z}{\color{blue}{y}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

      5. lower-/.f64 73.9

         \[\leadsto 0.5 \cdot \left(y - z \cdot \frac{z}{\color{blue}{y}}\right) \]

   8. Applied rewrites 73.9%

      \[\leadsto 0.5 \cdot \left(y - z \cdot \color{blue}{\frac{z}{y}}\right) \]

   if 4.00000000000000006e116 < x

   1. Initial program 60.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 85.6

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 85.6%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.5%

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{z \cdot z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+130}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x \cdot \frac{x}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (* z z) y_m)))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -4e-33)
      t_0
      (if (<= t_1 1e+130)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (* x (/ x y_m)) 0.5) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = -0.5 * ((z * z) / y_m);
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -4e-33) {
		tmp = t_0;
	} else if (t_1 <= 1e+130) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * (x / y_m)) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = -0.5 * ((z * z) / y_m);
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -4e-33) {
		tmp = t_0;
	} else if (t_1 <= 1e+130) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * (x / y_m)) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = -0.5 * ((z * z) / y_m)
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= -4e-33:
		tmp = t_0
	elif t_1 <= 1e+130:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (x * (x / y_m)) * 0.5
	else:
		tmp = t_0
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(-0.5 * Float64(Float64(z * z) / y_m))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -4e-33)
		tmp = t_0;
	elseif (t_1 <= 1e+130)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * Float64(x / y_m)) * 0.5);
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = -0.5 * ((z * z) / y_m);
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= -4e-33)
		tmp = t_0;
	elseif (t_1 <= 1e+130)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * (x / y_m)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -4e-33], t$95$0, If[LessEqual[t$95$1, 1e+130], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.0000000000000002e-33 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{{z}^{2}}{y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{{z}^{2}}{\color{blue}{y}} \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{z \cdot z}{y} \]

      4. lift-*.f64 79.0

         \[\leadsto -0.5 \cdot \frac{z \cdot z}{y} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z \cdot z}{y}} \]

   if -4.0000000000000002e-33 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.0000000000000001e130

   1. Initial program 93.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 69.4

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 69.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 1.0000000000000001e130 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 60.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 81.1

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 59.0

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 59.0%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.9%

       \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.0%

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;\left(x \cdot \frac{x}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1.85e+47) (* (* x (/ x y_m)) 0.5) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.85e+47) {
		tmp = (x * (x / y_m)) * 0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.85d+47) then
        tmp = (x * (x / y_m)) * 0.5d0
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.85e+47) {
		tmp = (x * (x / y_m)) * 0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.85e+47:
		tmp = (x * (x / y_m)) * 0.5
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.85e+47)
		tmp = Float64(Float64(x * Float64(x / y_m)) * 0.5);
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.85e+47)
		tmp = (x * (x / y_m)) * 0.5;
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.85e+47], N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000002e47

   1. Initial program 91.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \frac{1}{2}\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - {z}^{2}}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot x - z \cdot z}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, \frac{1}{2}, \frac{1}{2}\right) \cdot y \]

      14. lift-*.f64 84.9

          \[\leadsto \mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y \]

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x + z\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      8. lift--.f64 88.6

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   7. Applied rewrites 88.6%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.0%

       \[\leadsto \left(x \cdot \frac{x - z}{y}\right) \cdot 0.5 \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.9%

       \[\leadsto \left(x \cdot \frac{x}{y}\right) \cdot 0.5 \]

   if 1.8500000000000002e47 < y

   1. Initial program 40.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 64.4

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1.85e+47) (/ (* x x) (+ y_m y_m)) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.85e+47) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.85d+47) then
        tmp = (x * x) / (y_m + y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.85e+47) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.85e+47:
		tmp = (x * x) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.85e+47)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.85e+47)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.85e+47], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000002e47

   1. Initial program 91.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

      2. lift-*.f64 44.2

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

      4. lift-+.f64 44.2

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   6. Applied rewrites 44.2%

      \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   if 1.8500000000000002e47 < y

   1. Initial program 40.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 64.4

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* 0.5 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (0.5d0 * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (0.5 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (0.5 * y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.0%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.9

      \[\leadsto 0.5 \cdot \color{blue}{y} \]

4. Applied rewrites 34.9%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64
  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))