Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 10.38% → 4.11%

Time: 11.3s

Precision: binary64

Cost: 2124

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := \frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;t_1 \leq 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \end{array} \]

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))) (t_2 (/ (/ (* x 2.0) z) (- y t))))
   (if (<= t_1 -4e+220)
     t_2
     (if (<= t_1 -5e-254)
       (/ x (/ (* z (- y t)) 2.0))
       (if (<= t_1 1e-283) t_2 (* x (/ (/ 2.0 z) (- 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 t_2 = ((x * 2.0) / z) / (y - t);
	double tmp;
	if (t_1 <= -4e+220) {
		tmp = t_2;
	} else if (t_1 <= -5e-254) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else if (t_1 <= 1e-283) {
		tmp = t_2;
	} else {
		tmp = x * ((2.0 / z) / (y - t));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * z) - (z * t)
    t_2 = ((x * 2.0d0) / z) / (y - t)
    if (t_1 <= (-4d+220)) then
        tmp = t_2
    else if (t_1 <= (-5d-254)) then
        tmp = x / ((z * (y - t)) / 2.0d0)
    else if (t_1 <= 1d-283) then
        tmp = t_2
    else
        tmp = x * ((2.0d0 / z) / (y - t))
    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 t_2 = ((x * 2.0) / z) / (y - t);
	double tmp;
	if (t_1 <= -4e+220) {
		tmp = t_2;
	} else if (t_1 <= -5e-254) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else if (t_1 <= 1e-283) {
		tmp = t_2;
	} else {
		tmp = x * ((2.0 / z) / (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)
	t_2 = ((x * 2.0) / z) / (y - t)
	tmp = 0
	if t_1 <= -4e+220:
		tmp = t_2
	elif t_1 <= -5e-254:
		tmp = x / ((z * (y - t)) / 2.0)
	elif t_1 <= 1e-283:
		tmp = t_2
	else:
		tmp = x * ((2.0 / z) / (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))
	t_2 = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t))
	tmp = 0.0
	if (t_1 <= -4e+220)
		tmp = t_2;
	elseif (t_1 <= -5e-254)
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	elseif (t_1 <= 1e-283)
		tmp = t_2;
	else
		tmp = Float64(x * Float64(Float64(2.0 / z) / 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);
	t_2 = ((x * 2.0) / z) / (y - t);
	tmp = 0.0;
	if (t_1 <= -4e+220)
		tmp = t_2;
	elseif (t_1 <= -5e-254)
		tmp = x / ((z * (y - t)) / 2.0);
	elseif (t_1 <= 1e-283)
		tmp = t_2;
	else
		tmp = x * ((2.0 / z) / (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]}, Block[{t$95$2 = N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+220], t$95$2, If[LessEqual[t$95$1, -5e-254], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-283], t$95$2, N[(x * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	10.38%
Target	3.05%
Herbie	4.11%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4e220 or -5.0000000000000003e-254 < (-.f64 (*.f64 y z) (*.f64 t z)) < 9.99999999999999947e-284

   1. Initial program 23.6

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

   2. Simplified 0.62

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

      [Start] 23.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 23.6	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 0.62	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if -4e220 < (-.f64 (*.f64 y z) (*.f64 t z)) < -5.0000000000000003e-254

   1. Initial program 0.41

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

   2. Simplified 0.38

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

      [Start] 0.41	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 0.37	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 0.38	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

   if 9.99999999999999947e-284 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 11.54

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

   2. Simplified 7.8

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

      [Start] 11.54	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 11.64	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 8.36	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 7.8	\[ x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.11

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{-283}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.25%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-83} \lor \neg \left(x \leq 7 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 2
Error	28.18%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+36} \lor \neg \left(y \leq 4.7 \cdot 10^{+43}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 3
Error	27.95%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+37} \lor \neg \left(y \leq 4.2 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]

Alternative 4
Error	28.12%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 5
Error	27.98%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.45 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	27.86%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 7
Error	27.92%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 8
Error	5.74%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq -10000000:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 9
Error	8.43%
Cost	576

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

Alternative 10
Error	50.13%
Cost	448

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

Reproduce

herbie shell --seed 2023088 
(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))))