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

Percentage Accurate: 68.0% → 99.9%

Time: 3.4s

Alternatives: 7

Speedup: 1.2×

• 59.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 68.0% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (fma (/ (- x z) y) (+ z x) y) 0.5))

C

double code(double x, double y, double z) {
	return fma(((x - z) / y), (z + x), y) * 0.5;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(Float64(x - z) / y), Float64(z + x), y) * 0.5)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * N[(z + x), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. div-add N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

3. Applied rewrites 93.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 93.8%

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 93.8%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.8%

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

5. Applied rewrites 93.8%

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

7. Applied rewrites 99.9%

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

Alternative 2: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(\left|x\right| \cdot \left|x\right| + \left|y\right| \cdot \left|y\right|\right) - \left|z\right| \cdot \left|z\right|}{\left|y\right| \cdot 2}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(\left|z\right|, \frac{\left|z\right|}{-2 \cdot \left|y\right|}, \left|y\right| \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|x\right|}{\left|y\right|}, \left|z\right| + \left|x\right|, \left|y\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|x\right| - \left|z\right|}{\left|y\right|}, \left|z\right|, \left|y\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (-
           (+ (* (fabs x) (fabs x)) (* (fabs y) (fabs y)))
           (* (fabs z) (fabs z)))
          (* (fabs y) 2.0))))
   (*
    (copysign 1.0 y)
    (if (<= t_0 -2e-38)
      (fma (fabs z) (/ (fabs z) (* -2.0 (fabs y))) (* (fabs y) 0.5))
      (if (<= t_0 INFINITY)
        (* (fma (/ (fabs x) (fabs y)) (+ (fabs z) (fabs x)) (fabs y)) 0.5)
        (* (fma (/ (- (fabs x) (fabs z)) (fabs y)) (fabs z) (fabs y)) 0.5))))))

C

double code(double x, double y, double z) {
	double t_0 = (((fabs(x) * fabs(x)) + (fabs(y) * fabs(y))) - (fabs(z) * fabs(z))) / (fabs(y) * 2.0);
	double tmp;
	if (t_0 <= -2e-38) {
		tmp = fma(fabs(z), (fabs(z) / (-2.0 * fabs(y))), (fabs(y) * 0.5));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((fabs(x) / fabs(y)), (fabs(z) + fabs(x)), fabs(y)) * 0.5;
	} else {
		tmp = fma(((fabs(x) - fabs(z)) / fabs(y)), fabs(z), fabs(y)) * 0.5;
	}
	return copysign(1.0, y) * tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(abs(x) * abs(x)) + Float64(abs(y) * abs(y))) - Float64(abs(z) * abs(z))) / Float64(abs(y) * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-38)
		tmp = fma(abs(z), Float64(abs(z) / Float64(-2.0 * abs(y))), Float64(abs(y) * 0.5));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(abs(x) / abs(y)), Float64(abs(z) + abs(x)), abs(y)) * 0.5);
	else
		tmp = Float64(fma(Float64(Float64(abs(x) - abs(z)) / abs(y)), abs(z), abs(y)) * 0.5);
	end
	return Float64(copysign(1.0, y) * tmp)
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -2e-38], N[(N[Abs[z], $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] / N[(-2.0 * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Abs[x], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]), $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))) < -1.9999999999999999e-38

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. div-add N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-frac-neg N/A

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

      12. frac-2neg N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{z}{-2 \cdot y}, \left(y + \frac{x \cdot x}{y}\right) \cdot 0.5\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z, \frac{z}{-2 \cdot y}, \color{blue}{y} \cdot 0.5\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(z, \frac{z}{-2 \cdot y}, \color{blue}{y} \cdot 0.5\right) \]

   if -1.9999999999999999e-38 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.8%

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8%

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

   5. Applied rewrites 93.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, z + x, y\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lower-/.f64 71.6%

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

   10. Applied rewrites 71.6%

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

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

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.8%

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8%

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

   5. Applied rewrites 93.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x - z}{y}, \color{blue}{z}, y\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

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

Alternative 3: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(\left|x\right| \cdot \left|x\right| + \left|y\right| \cdot \left|y\right|\right) - \left|z\right| \cdot \left|z\right|}{\left|y\right| \cdot 2}\\ t_1 := \frac{\left|x\right| - \left|z\right|}{\left|y\right|}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\left(\left|z\right| \cdot t\_1\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|x\right|}{\left|y\right|}, \left|z\right| + \left|x\right|, \left|y\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|z\right|, \left|y\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (-
           (+ (* (fabs x) (fabs x)) (* (fabs y) (fabs y)))
           (* (fabs z) (fabs z)))
          (* (fabs y) 2.0)))
        (t_1 (/ (- (fabs x) (fabs z)) (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= t_0 -2e-38)
      (* (* (fabs z) t_1) 0.5)
      (if (<= t_0 INFINITY)
        (* (fma (/ (fabs x) (fabs y)) (+ (fabs z) (fabs x)) (fabs y)) 0.5)
        (* (fma t_1 (fabs z) (fabs y)) 0.5))))))

C

double code(double x, double y, double z) {
	double t_0 = (((fabs(x) * fabs(x)) + (fabs(y) * fabs(y))) - (fabs(z) * fabs(z))) / (fabs(y) * 2.0);
	double t_1 = (fabs(x) - fabs(z)) / fabs(y);
	double tmp;
	if (t_0 <= -2e-38) {
		tmp = (fabs(z) * t_1) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((fabs(x) / fabs(y)), (fabs(z) + fabs(x)), fabs(y)) * 0.5;
	} else {
		tmp = fma(t_1, fabs(z), fabs(y)) * 0.5;
	}
	return copysign(1.0, y) * tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(abs(x) * abs(x)) + Float64(abs(y) * abs(y))) - Float64(abs(z) * abs(z))) / Float64(abs(y) * 2.0))
	t_1 = Float64(Float64(abs(x) - abs(z)) / abs(y))
	tmp = 0.0
	if (t_0 <= -2e-38)
		tmp = Float64(Float64(abs(z) * t_1) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(abs(x) / abs(y)), Float64(abs(z) + abs(x)), abs(y)) * 0.5);
	else
		tmp = Float64(fma(t_1, abs(z), abs(y)) * 0.5);
	end
	return Float64(copysign(1.0, y) * tmp)
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -2e-38], N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Abs[x], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$1 * N[Abs[z], $MachinePrecision] + N[Abs[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]), $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))) < -1.9999999999999999e-38

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval 73.2%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.0%

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 35.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.2%

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lift-/.f64 N/A

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

       8. lower-*.f64 39.6%

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

   11. Applied rewrites 39.6%

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

   if -1.9999999999999999e-38 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.8%

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8%

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

   5. Applied rewrites 93.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, z + x, y\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lower-/.f64 71.6%

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

   10. Applied rewrites 71.6%

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

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

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.8%

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8%

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

   5. Applied rewrites 93.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x - z}{y}, \color{blue}{z}, y\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

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

Alternative 4: 79.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (* (fma (/ (- (fabs x) (fabs z)) y) (fabs z) y) 0.5))

C

double code(double x, double y, double z) {
	return fma(((fabs(x) - fabs(z)) / y), fabs(z), y) * 0.5;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(Float64(abs(x) - abs(z)) / y), abs(z), y) * 0.5)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. div-add N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

3. Applied rewrites 93.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 93.8%

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 93.8%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.8%

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

5. Applied rewrites 93.8%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{x - z}{y}, \color{blue}{z}, y\right) \cdot 0.5 \]
9. Step-by-step derivation

10. Applied rewrites 72.9%

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

Alternative 5: 64.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\left|y\right| \leq 3.1 \cdot 10^{+104}:\\ \;\;\;\;\left(\left|z\right| \cdot \frac{\left|x\right| - \left|z\right|}{\left|y\right|}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (*
  (copysign 1.0 y)
  (if (<= (fabs y) 3.1e+104)
    (* (* (fabs z) (/ (- (fabs x) (fabs z)) (fabs y))) 0.5)
    (* 0.5 (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 3.1e+104) {
		tmp = (fabs(z) * ((fabs(x) - fabs(z)) / fabs(y))) * 0.5;
	} else {
		tmp = 0.5 * fabs(y);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 3.1e+104) {
		tmp = (Math.abs(z) * ((Math.abs(x) - Math.abs(z)) / Math.abs(y))) * 0.5;
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 3.1e+104:
		tmp = (math.fabs(z) * ((math.fabs(x) - math.fabs(z)) / math.fabs(y))) * 0.5
	else:
		tmp = 0.5 * math.fabs(y)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 3.1e+104)
		tmp = Float64(Float64(abs(z) * Float64(Float64(abs(x) - abs(z)) / abs(y))) * 0.5);
	else
		tmp = Float64(0.5 * abs(y));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 3.1e+104)
		tmp = (abs(z) * ((abs(x) - abs(z)) / abs(y))) * 0.5;
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[y], $MachinePrecision], 3.1e+104], N[(N[(N[Abs[z], $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000017e104

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval 73.2%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.0%

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 35.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.2%

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lift-/.f64 N/A

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

       8. lower-*.f64 39.6%

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

   11. Applied rewrites 39.6%

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

   if 3.10000000000000017e104 < y

   1. Initial program 68.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.7%

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

   4. Applied rewrites 34.7%

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

Alternative 6: 60.0% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\left|y\right| \leq 6.4 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \frac{\left|z\right| \cdot \left(\left|x\right| - \left|z\right|\right)}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (*
  (copysign 1.0 y)
  (if (<= (fabs y) 6.4e+105)
    (* 0.5 (/ (* (fabs z) (- (fabs x) (fabs z))) (fabs y)))
    (* 0.5 (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 6.4e+105) {
		tmp = 0.5 * ((fabs(z) * (fabs(x) - fabs(z))) / fabs(y));
	} else {
		tmp = 0.5 * fabs(y);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 6.4e+105) {
		tmp = 0.5 * ((Math.abs(z) * (Math.abs(x) - Math.abs(z))) / Math.abs(y));
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 6.4e+105:
		tmp = 0.5 * ((math.fabs(z) * (math.fabs(x) - math.fabs(z))) / math.fabs(y))
	else:
		tmp = 0.5 * math.fabs(y)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 6.4e+105)
		tmp = Float64(0.5 * Float64(Float64(abs(z) * Float64(abs(x) - abs(z))) / abs(y)));
	else
		tmp = Float64(0.5 * abs(y));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 6.4e+105)
		tmp = 0.5 * ((abs(z) * (abs(x) - abs(z))) / abs(y));
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[y], $MachinePrecision], 6.4e+105], N[(0.5 * N[(N[(N[Abs[z], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e105

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval 73.2%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.0%

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 35.2%

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

   if 6.4e105 < y

   1. Initial program 68.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.7%

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

   4. Applied rewrites 34.7%

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

Alternative 7: 34.7% accurate, 5.4× speedup

Math

\[0.5 \cdot y \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

double code(double x, double y, double z) {
	return 0.5 * y;
}

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 = 0.5d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

def code(x, y, z):
	return 0.5 * y

Julia

function code(x, y, z)
	return Float64(0.5 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * y;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.7%

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

4. Applied rewrites 34.7%

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

Reproduce

herbie shell --seed 2025183 
(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)))