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

Percentage Accurate: 68.7% → 96.6%

Time: 4.7s

Alternatives: 16

Speedup: 1.2×

• 41.9% 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: 68.7% 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: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 (* (+ z x) (/ (- x z) y_m))))
   (*
    y_s
    (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e+99)
      (* 0.5 t_0)
      (* (fma (/ t_0 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 t_0 = (z + x) * ((x - z) / y_m);
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e+99) {
		tmp = 0.5 * t_0;
	} else {
		tmp = fma((t_0 / y_m), 0.5, 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)
	t_0 = Float64(Float64(z + x) * Float64(Float64(x - z) / y_m))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e+99)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(fma(Float64(t_0 / y_m), 0.5, 0.5) * 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_] := Block[{t$95$0 = N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[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], -2e+99], N[(0.5 * t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $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))) < -1.9999999999999999e99

   1. Initial program 74.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 68.4

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

   8. Applied rewrites 68.4%

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

   if -1.9999999999999999e99 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 60.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 84.0%

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 96.4

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

   7. Applied rewrites 96.4%

      \[\leadsto \mathsf{fma}\left(\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{y}, 0.5, 0.5\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{y}, 0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z \cdot \frac{z}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* z (/ z y_m)) -0.5))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -5e-105)
      t_0
      (if (<= t_1 1e+151)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* 0.5 (* (+ z x) (/ x 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 = (z * (z / y_m)) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} 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 = (z * (z / y_m)) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} 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 = (z * (z / y_m)) * -0.5
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= -5e-105:
		tmp = t_0
	elif t_1 <= 1e+151:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = 0.5 * ((z + x) * (x / 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(Float64(z * Float64(z / y_m)) * -0.5)
	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 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(x / y_m)));
	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 = (z * (z / y_m)) * -0.5;
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = 0.5 * ((z + x) * (x / y_m));
	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[(N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $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, -5e-105], t$95$0, If[LessEqual[t$95$1, 1e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -4.99999999999999963e-105 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 37.1

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

   8. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{z \cdot z}{y} \cdot -0.5} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 39.5

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

   10. Applied rewrites 39.5%

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

   if -4.99999999999999963e-105 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000002e151

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if 1.00000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 65.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 87.1%

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 96.0

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. difference-of-squares-rev N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 69.1

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

   10. Applied rewrites 69.1%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 37.8%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z \cdot \frac{z}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* z (/ z y_m)) -0.5))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -5e-105)
      t_0
      (if (<= t_1 1e+151)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* 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 = (z * (z / y_m)) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} 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 = (z * (z / y_m)) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} 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 = (z * (z / y_m)) * -0.5
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= -5e-105:
		tmp = t_0
	elif t_1 <= 1e+151:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (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(Float64(z * Float64(z / y_m)) * -0.5)
	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 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	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 = (z * (z / y_m)) * -0.5;
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	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[(N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $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, -5e-105], t$95$0, If[LessEqual[t$95$1, 1e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $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))) < -4.99999999999999963e-105 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 37.1

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

   8. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{z \cdot z}{y} \cdot -0.5} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 39.5

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

   10. Applied rewrites 39.5%

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

   if -4.99999999999999963e-105 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000002e151

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if 1.00000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 65.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} - {z}^{2}}}{y \cdot 2} \]
   4. 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 63.7

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

   5. Applied rewrites 63.7%

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 33.4

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

   10. Applied rewrites 33.4%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{y + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{z \cdot z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (/ (* z z) y_m)))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -5e-105)
      t_0
      (if (<= t_1 1e+151)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* 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 * ((z * z) / y_m);
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} 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 <= -5e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} 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 <= -5e-105:
		tmp = t_0
	elif t_1 <= 1e+151:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (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(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 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	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 <= -5e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+151)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	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, -5e-105], t$95$0, If[LessEqual[t$95$1, 1e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $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))) < -4.99999999999999963e-105 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   if -4.99999999999999963e-105 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000002e151

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if 1.00000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 65.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} - {z}^{2}}}{y \cdot 2} \]
   4. 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 63.7

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

   5. Applied rewrites 63.7%

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 33.4

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

   10. Applied rewrites 33.4%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{y + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ t_1 := \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot t\_1\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(z + x\right) \cdot \frac{x}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot t\_1}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)))
        (t_1 (/ (- x z) y_m)))
   (*
    y_s
    (if (<= t_0 -5e-105)
      (* 0.5 (* (+ z x) t_1))
      (if (<= t_0 INFINITY)
        (* (fma (/ (* (+ z x) (/ x y_m)) y_m) 0.5 0.5) y_m)
        (* (fma (/ (* z t_1) 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 t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double t_1 = (x - z) / y_m;
	double tmp;
	if (t_0 <= -5e-105) {
		tmp = 0.5 * ((z + x) * t_1);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((((z + x) * (x / y_m)) / y_m), 0.5, 0.5) * y_m;
	} else {
		tmp = fma(((z * t_1) / y_m), 0.5, 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)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	t_1 = Float64(Float64(x - z) / y_m)
	tmp = 0.0
	if (t_0 <= -5e-105)
		tmp = Float64(0.5 * Float64(Float64(z + x) * t_1));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(Float64(Float64(z + x) * Float64(x / y_m)) / y_m), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(Float64(z * t_1) / y_m), 0.5, 0.5) * 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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-105], N[(0.5 * N[(N[(z + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(z * t$95$1), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $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))) < -4.99999999999999963e-105

   1. Initial program 78.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 65.4

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

   8. Applied rewrites 65.4%

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

   if -4.99999999999999963e-105 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 87.2%

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 73.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 67.6%

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 85.4

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

   10. Applied rewrites 85.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(z + x\right) \cdot \frac{x}{y}}{y}, 0.5, 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \frac{x - z}{y}}{y}, 0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot t\_0\right)\\ \mathbf{elif}\;y\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 (/ (- x z) y_m)))
   (*
    y_s
    (if (<= y_m 1.05e-135)
      (* 0.5 (* (+ z x) t_0))
      (if (<= y_m 1.35e+154)
        (* (fma (* (+ z x) (/ (- x z) (* y_m y_m))) 0.5 0.5) y_m)
        (* (fma (/ (* z t_0) 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 t_0 = (x - z) / y_m;
	double tmp;
	if (y_m <= 1.05e-135) {
		tmp = 0.5 * ((z + x) * t_0);
	} else if (y_m <= 1.35e+154) {
		tmp = fma(((z + x) * ((x - z) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma(((z * t_0) / y_m), 0.5, 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)
	t_0 = Float64(Float64(x - z) / y_m)
	tmp = 0.0
	if (y_m <= 1.05e-135)
		tmp = Float64(0.5 * Float64(Float64(z + x) * t_0));
	elseif (y_m <= 1.35e+154)
		tmp = Float64(fma(Float64(Float64(z + x) * Float64(Float64(x - z) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(Float64(z * t_0) / y_m), 0.5, 0.5) * 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_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.05e-135], N[(0.5 * N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.35e+154], N[(N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(z * t$95$0), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-135

   1. Initial program 67.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 72.1

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

   8. Applied rewrites 72.1%

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

   if 1.05e-135 < y < 1.35000000000000003e154

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 95.5%

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

      1. associate-/l/ N/A

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

      2. pow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

   if 1.35000000000000003e154 < y

   1. Initial program 10.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 70.9%

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y \cdot y}, 0.5, 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \frac{x - z}{y}}{y}, 0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{y\_m} \cdot z}{y\_m}, -0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 1.05e-135)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 1.35e+154)
      (* (fma (* (+ z x) (/ (- x z) (* y_m y_m))) 0.5 0.5) y_m)
      (* (fma (/ (* (/ z y_m) z) 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 <= 1.05e-135) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 1.35e+154) {
		tmp = fma(((z + x) * ((x - z) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma((((z / y_m) * z) / y_m), -0.5, 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 <= 1.05e-135)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 1.35e+154)
		tmp = Float64(fma(Float64(Float64(z + x) * Float64(Float64(x - z) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(z / y_m) * z) / y_m), -0.5, 0.5) * 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, 1.05e-135], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.35e+154], N[(N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] / y$95$m), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-135

   1. Initial program 67.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 72.1

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

   8. Applied rewrites 72.1%

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

   if 1.05e-135 < y < 1.35000000000000003e154

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 95.5%

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

      1. associate-/l/ N/A

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

      2. pow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

   if 1.35000000000000003e154 < y

   1. Initial program 10.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.8

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

   10. Applied rewrites 88.8%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 88.9

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

   12. Applied rewrites 88.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y \cdot y}, 0.5, 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{y} \cdot z}{y}, -0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{y\_m} \cdot z}{y\_m}, -0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 8e-140)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 3.6e+147)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* (fma (/ (* (/ z y_m) z) 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 <= 8e-140) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 3.6e+147) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = fma((((z / y_m) * z) / y_m), -0.5, 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 <= 8e-140)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 3.6e+147)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(z / y_m) * z) / y_m), -0.5, 0.5) * 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, 8e-140], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.6e+147], 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[(N[(N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] / y$95$m), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 7.9999999999999999e-140

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 7.9999999999999999e-140 < y < 3.6000000000000002e147

   1. Initial program 91.8%

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

   if 3.6000000000000002e147 < y

   1. Initial program 10.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.1

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

   10. Applied rewrites 89.1%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 89.2

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

   12. Applied rewrites 89.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{y} \cdot z}{y}, -0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y\_m} \cdot \frac{z}{y\_m}, -0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 8e-140)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 3.6e+147)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* (fma (* (/ z y_m) (/ z 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 <= 8e-140) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 3.6e+147) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = fma(((z / y_m) * (z / y_m)), -0.5, 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 <= 8e-140)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 3.6e+147)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(fma(Float64(Float64(z / y_m) * Float64(z / y_m)), -0.5, 0.5) * 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, 8e-140], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.6e+147], 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[(N[(N[(N[(z / y$95$m), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 7.9999999999999999e-140

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 7.9999999999999999e-140 < y < 3.6000000000000002e147

   1. Initial program 91.8%

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

   if 3.6000000000000002e147 < y

   1. Initial program 10.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.1

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

   10. Applied rewrites 89.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y} \cdot \frac{z}{y}, -0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{y\_m \cdot y\_m}, -0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 8e-140)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 1.55e+52)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* (fma (* z (/ z (* y_m 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 <= 8e-140) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 1.55e+52) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = fma((z * (z / (y_m * y_m))), -0.5, 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 <= 8e-140)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 1.55e+52)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(fma(Float64(z * Float64(z / Float64(y_m * y_m))), -0.5, 0.5) * 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, 8e-140], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.55e+52], 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[(N[(N[(z * N[(z / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 7.9999999999999999e-140

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 7.9999999999999999e-140 < y < 1.55e52

   1. Initial program 97.1%

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

   if 1.55e52 < y

   1. Initial program 41.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 82.1

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

   10. Applied rewrites 82.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{y \cdot y}, -0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(y\_m + z\right) \cdot \left(y\_m - z\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 6.2e+22)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 1.4e+158)
      (/ (* (+ y_m z) (- y_m z)) (* y_m 2.0))
      (* 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 <= 6.2e+22) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 1.4e+158) {
		tmp = ((y_m + z) * (y_m - z)) / (y_m * 2.0);
	} 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 <= 6.2d+22) then
        tmp = 0.5d0 * ((z + x) * ((x - z) / y_m))
    else if (y_m <= 1.4d+158) then
        tmp = ((y_m + z) * (y_m - z)) / (y_m * 2.0d0)
    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 <= 6.2e+22) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 1.4e+158) {
		tmp = ((y_m + z) * (y_m - z)) / (y_m * 2.0);
	} 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 <= 6.2e+22:
		tmp = 0.5 * ((z + x) * ((x - z) / y_m))
	elif y_m <= 1.4e+158:
		tmp = ((y_m + z) * (y_m - z)) / (y_m * 2.0)
	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 <= 6.2e+22)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 1.4e+158)
		tmp = Float64(Float64(Float64(y_m + z) * Float64(y_m - z)) / Float64(y_m * 2.0));
	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 <= 6.2e+22)
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	elseif (y_m <= 1.4e+158)
		tmp = ((y_m + z) * (y_m - z)) / (y_m * 2.0);
	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, 6.2e+22], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.4e+158], N[(N[(N[(y$95$m + z), $MachinePrecision] * N[(y$95$m - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 6.2000000000000004e22

   1. Initial program 72.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if 6.2000000000000004e22 < y < 1.40000000000000001e158

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if 1.40000000000000001e158 < y

   1. Initial program 8.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(y + z\right) \cdot \left(y - z\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{y\_m \cdot y\_m}, -0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 6.2e+22)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (* (fma (* z (/ z (* y_m 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 <= 6.2e+22) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else {
		tmp = fma((z * (z / (y_m * y_m))), -0.5, 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 <= 6.2e+22)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	else
		tmp = Float64(fma(Float64(z * Float64(z / Float64(y_m * y_m))), -0.5, 0.5) * 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, 6.2e+22], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(z / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000004e22

   1. Initial program 72.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if 6.2000000000000004e22 < y

   1. Initial program 45.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 83.2

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

   10. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{z}{y \cdot y}, -0.5, 0.5\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{y \cdot y}, -0.5, 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 5.8e+192) (* 0.5 (* (+ z x) (/ (- x z) 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 <= 5.8e+192) {
		tmp = 0.5 * ((z + x) * ((x - z) / 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 <= 5.8d+192) then
        tmp = 0.5d0 * ((z + x) * ((x - z) / 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 <= 5.8e+192) {
		tmp = 0.5 * ((z + x) * ((x - z) / 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 <= 5.8e+192:
		tmp = 0.5 * ((z + x) * ((x - z) / 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 <= 5.8e+192)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / 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 <= 5.8e+192)
		tmp = 0.5 * ((z + x) * ((x - z) / 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, 5.8e+192], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.8000000000000003e192

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. 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

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares-rev N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if 5.8000000000000003e192 < y

   1. Initial program 5.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \left(x - z\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 4.4e+102) (/ (* (+ x z) (- x z)) (+ 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 <= 4.4e+102) {
		tmp = ((x + z) * (x - z)) / (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 <= 4.4d+102) then
        tmp = ((x + z) * (x - z)) / (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 <= 4.4e+102) {
		tmp = ((x + z) * (x - z)) / (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 <= 4.4e+102:
		tmp = ((x + z) * (x - z)) / (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 <= 4.4e+102)
		tmp = Float64(Float64(Float64(x + z) * Float64(x - z)) / 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 <= 4.4e+102)
		tmp = ((x + z) * (x - z)) / (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, 4.4e+102], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $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 < 4.40000000000000015e102

   1. Initial program 73.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} - {z}^{2}}}{y \cdot 2} \]
   4. 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 68.7

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

   5. Applied rewrites 68.7%

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 68.7

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

   7. Applied rewrites 68.7%

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

   if 4.40000000000000015e102 < y

   1. Initial program 33.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \left(x - z\right)}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \end{array} \end{array} \]

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 (<= x 3.4e+89) (* 0.5 y_m) (/ (* x x) (+ y_m 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 (x <= 3.4e+89) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + 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 (x <= 3.4d+89) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x * x) / (y_m + 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 (x <= 3.4e+89) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + 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 x <= 3.4e+89:
		tmp = 0.5 * y_m
	else:
		tmp = (x * x) / (y_m + 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 (x <= 3.4e+89)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x * x) / Float64(y_m + 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 (x <= 3.4e+89)
		tmp = 0.5 * y_m;
	else
		tmp = (x * x) / (y_m + 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[x, 3.4e+89], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.4000000000000002e89

   1. Initial program 66.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   if 3.4000000000000002e89 < x

   1. Initial program 61.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} - {z}^{2}}}{y \cdot 2} \]
   4. 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 81.2

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

   5. Applied rewrites 81.2%

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

      1. *-commutative N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 81.2

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

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 53.9

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

   10. Applied rewrites 53.9%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{y + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.4% accurate, 6.3× 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 65.7%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 36.0

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

5. Applied rewrites 36.0%

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

6. Final simplification 36.0%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

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

C

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

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 = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025044 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))