Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.5 → 1.2

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{2 \cdot x}{y - t} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 7.93994715672202 \cdot 10^{+280}:\\ \;\;\;\;\frac{2 \cdot x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \frac{2}{y - t}\right)\\ \end{array} \]

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 (- INFINITY))
     (* (/ (* 2.0 x) (- y t)) (/ 1.0 z))
     (if (<= t_1 7.93994715672202e+280)
       (/ (* 2.0 x) t_1)
       (* (/ 1.0 z) (* x (/ 2.0 (- y t))))))))

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 <= -((double) INFINITY)) {
		tmp = ((2.0 * x) / (y - t)) * (1.0 / z);
	} else if (t_1 <= 7.93994715672202e+280) {
		tmp = (2.0 * x) / t_1;
	} else {
		tmp = (1.0 / z) * (x * (2.0 / (y - t)));
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = ((2.0 * x) / (y - t)) * (1.0 / z);
	} else if (t_1 <= 7.93994715672202e+280) {
		tmp = (2.0 * x) / t_1;
	} else {
		tmp = (1.0 / z) * (x * (2.0 / (y - t)));
	}
	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 <= -math.inf:
		tmp = ((2.0 * x) / (y - t)) * (1.0 / z)
	elif t_1 <= 7.93994715672202e+280:
		tmp = (2.0 * x) / t_1
	else:
		tmp = (1.0 / z) * (x * (2.0 / (y - t)))
	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 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(2.0 * x) / Float64(y - t)) * Float64(1.0 / z));
	elseif (t_1 <= 7.93994715672202e+280)
		tmp = Float64(Float64(2.0 * x) / t_1);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(x * Float64(2.0 / Float64(y - t))));
	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 <= -Inf)
		tmp = ((2.0 * x) / (y - t)) * (1.0 / z);
	elseif (t_1 <= 7.93994715672202e+280)
		tmp = (2.0 * x) / t_1;
	else
		tmp = (1.0 / z) * (x * (2.0 / (y - t)));
	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, (-Infinity)], N[(N[(N[(2.0 * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 7.93994715672202e+280], N[(N[(2.0 * x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(x * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.1
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 20.4

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

   2. Simplified 18.4

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

   3. Applied *-un-lft-identity_binary64 18.4

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

   4. Applied *-un-lft-identity_binary64 18.4

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

   5. Applied add-sqr-sqrt_binary64 18.4

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

   6. Applied times-frac_binary64 18.4

      \[\leadsto x \cdot \frac{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{y - t}}}{1 \cdot z} \]

   7. Applied times-frac_binary64 18.4

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

   8. Applied associate-*r*_binary64 18.4

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

   9. Simplified 18.4

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

   10. Applied div-inv_binary64 18.4

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

   11. Applied associate-*r*_binary64 0.4

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

   12. Simplified 0.2

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 7.93994715672202011e280

   1. Initial program 1.6

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

   if 7.93994715672202011e280 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 23.2

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

   2. Simplified 14.9

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

   3. Applied div-inv_binary64 14.9

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

   4. Applied associate-*r*_binary64 0.2

      \[\leadsto \color{blue}{\left(x \cdot \frac{2}{y - t}\right) \cdot \frac{1}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{2 \cdot x}{y - t} \cdot \frac{1}{z}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 7.93994715672202 \cdot 10^{+280}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \frac{2}{y - t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))