Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.2% → 95.4%

Time: 14.8s

Precision: binary64

Cost: 2125

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-173} \lor \neg \left(t_1 \leq 5 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) (* z t))))
   (if (<= t_1 -1e-163)
     (* x (/ (/ 2.0 z) (- y t)))
     (if (or (<= t_1 2e-173) (not (<= t_1 5e+136)))
       (/ (/ (* x 2.0) z) (- y t))
       (/ x (/ (* z (- y t)) 2.0))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double tmp;
	if (t_1 <= -1e-163) {
		tmp = x * ((2.0 / z) / (y - t));
	} else if ((t_1 <= 2e-173) || !(t_1 <= 5e+136)) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z) - (z * t)
    if (t_1 <= (-1d-163)) then
        tmp = x * ((2.0d0 / z) / (y - t))
    else if ((t_1 <= 2d-173) .or. (.not. (t_1 <= 5d+136))) then
        tmp = ((x * 2.0d0) / z) / (y - t)
    else
        tmp = x / ((z * (y - t)) / 2.0d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double tmp;
	if (t_1 <= -1e-163) {
		tmp = x * ((2.0 / z) / (y - t));
	} else if ((t_1 <= 2e-173) || !(t_1 <= 5e+136)) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (y * z) - (z * t)
	tmp = 0
	if t_1 <= -1e-163:
		tmp = x * ((2.0 / z) / (y - t))
	elif (t_1 <= 2e-173) or not (t_1 <= 5e+136):
		tmp = ((x * 2.0) / z) / (y - t)
	else:
		tmp = x / ((z * (y - t)) / 2.0)
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e-163)
		tmp = Float64(x * Float64(Float64(2.0 / z) / Float64(y - t)));
	elseif ((t_1 <= 2e-173) || !(t_1 <= 5e+136))
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	else
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - (z * t);
	tmp = 0.0;
	if (t_1 <= -1e-163)
		tmp = x * ((2.0 / z) / (y - t));
	elseif ((t_1 <= 2e-173) || ~((t_1 <= 5e+136)))
		tmp = ((x * 2.0) / z) / (y - t);
	else
		tmp = x / ((z * (y - t)) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-163], N[(x * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-173], N[Not[LessEqual[t$95$1, 5e+136]], $MachinePrecision]], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	89.2%
Target	96.7%
Herbie	95.4%

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -9.99999999999999923e-164

   1. Initial program 91.9%

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

   2. Simplified 92.4%

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

      [Start] 91.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 91.9	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 91.9	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 92.4	\[ x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}} \]

   if -9.99999999999999923e-164 < (-.f64 (*.f64 y z) (*.f64 t z)) < 2.0000000000000001e-173 or 5.0000000000000002e136 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 78.8%

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

   2. Simplified 96.7%

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

      [Start] 78.8	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 84.3	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 96.7	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if 2.0000000000000001e-173 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000002e136

   1. Initial program 99.5%

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

   2. Simplified 99.6%

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

      [Start] 99.5	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 99.6	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 2 \cdot 10^{-173} \lor \neg \left(y \cdot z - z \cdot t \leq 5 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.8%
Cost	1372

\[\begin{array}{l} t_1 := \frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq 880:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 2
Accuracy	89.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-216} \lor \neg \left(t \leq 10^{-182}\right):\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \end{array} \]

Alternative 3
Accuracy	90.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-237} \lor \neg \left(t \leq 1.2 \cdot 10^{-176}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	90.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-237}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 5
Accuracy	90.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -5000000000:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \end{array} \]

Alternative 6
Accuracy	93.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -0.0001:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \end{array} \]

Alternative 7
Accuracy	73.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-31} \lor \neg \left(y \leq 1.85 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 8
Accuracy	73.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-31} \lor \neg \left(y \leq 2.2 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 9
Accuracy	73.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-31} \lor \neg \left(y \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 10
Accuracy	73.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-31} \lor \neg \left(y \leq 6.3 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 11
Accuracy	73.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-31} \lor \neg \left(y \leq 4.8 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z}\\ \end{array} \]

Alternative 12
Accuracy	50.5%
Cost	448

\[x \cdot \frac{-2}{z \cdot t} \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

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