Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 0.7

Time: 7.9s

Precision: binary64

Cost: 2640

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{y - t}}{-z}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-207}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t_1 \leq 10^{+236}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

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 (/ (* z (- y t)) 2.0))))
   (if (<= t_1 -1e+249)
     (/ (/ (* x -2.0) (- y t)) (- z))
     (if (<= t_1 -4e-77)
       t_2
       (if (<= t_1 1e-207)
         (* 2.0 (/ (/ x z) (- y t)))
         (if (<= t_1 1e+236) t_2 (* (/ x z) (/ 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 t_2 = x / ((z * (y - t)) / 2.0);
	double tmp;
	if (t_1 <= -1e+249) {
		tmp = ((x * -2.0) / (y - t)) / -z;
	} else if (t_1 <= -4e-77) {
		tmp = t_2;
	} else if (t_1 <= 1e-207) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 1e+236) {
		tmp = t_2;
	} else {
		tmp = (x / z) * (2.0 / (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 / ((z * (y - t)) / 2.0d0)
    if (t_1 <= (-1d+249)) then
        tmp = ((x * (-2.0d0)) / (y - t)) / -z
    else if (t_1 <= (-4d-77)) then
        tmp = t_2
    else if (t_1 <= 1d-207) then
        tmp = 2.0d0 * ((x / z) / (y - t))
    else if (t_1 <= 1d+236) then
        tmp = t_2
    else
        tmp = (x / z) * (2.0d0 / (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 / ((z * (y - t)) / 2.0);
	double tmp;
	if (t_1 <= -1e+249) {
		tmp = ((x * -2.0) / (y - t)) / -z;
	} else if (t_1 <= -4e-77) {
		tmp = t_2;
	} else if (t_1 <= 1e-207) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 1e+236) {
		tmp = t_2;
	} else {
		tmp = (x / z) * (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)
	t_2 = x / ((z * (y - t)) / 2.0)
	tmp = 0
	if t_1 <= -1e+249:
		tmp = ((x * -2.0) / (y - t)) / -z
	elif t_1 <= -4e-77:
		tmp = t_2
	elif t_1 <= 1e-207:
		tmp = 2.0 * ((x / z) / (y - t))
	elif t_1 <= 1e+236:
		tmp = t_2
	else:
		tmp = (x / z) * (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))
	t_2 = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0))
	tmp = 0.0
	if (t_1 <= -1e+249)
		tmp = Float64(Float64(Float64(x * -2.0) / Float64(y - t)) / Float64(-z));
	elseif (t_1 <= -4e-77)
		tmp = t_2;
	elseif (t_1 <= 1e-207)
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	elseif (t_1 <= 1e+236)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / z) * 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);
	t_2 = x / ((z * (y - t)) / 2.0);
	tmp = 0.0;
	if (t_1 <= -1e+249)
		tmp = ((x * -2.0) / (y - t)) / -z;
	elseif (t_1 <= -4e-77)
		tmp = t_2;
	elseif (t_1 <= 1e-207)
		tmp = 2.0 * ((x / z) / (y - t));
	elseif (t_1 <= 1e+236)
		tmp = t_2;
	else
		tmp = (x / z) * (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]}, Block[{t$95$2 = N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+249], N[(N[(N[(x * -2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[t$95$1, -4e-77], t$95$2, If[LessEqual[t$95$1, 1e-207], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+236], t$95$2, N[(N[(x / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	6.8
Target	2.1
Herbie	0.7

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -9.9999999999999992e248

   1. Initial program 15.2

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

   2. Simplified 15.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}} \]
      Proof
      (/.f64 x (/.f64 (*.f64 z (-.f64 y t)) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))) 2)): 5 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 15.2

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{y - t}}{-z}} \]
      Proof
      (/.f64 (/.f64 (*.f64 x -2) (-.f64 y t)) (neg.f64 z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 x -2) (*.f64 (-.f64 y t) (neg.f64 z)))): 53 points increase in error, 39 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x (*.f64 (-.f64 y t) (neg.f64 z))) -2)): 0 points increase in error, 0 points decrease in error

   if -9.9999999999999992e248 < (-.f64 (*.f64 y z) (*.f64 t z)) < -3.9999999999999997e-77 or 9.99999999999999925e-208 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.00000000000000005e236

   1. Initial program 0.3

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}} \]
      Proof
      (/.f64 x (/.f64 (*.f64 z (-.f64 y t)) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))) 2)): 5 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   if -3.9999999999999997e-77 < (-.f64 (*.f64 y z) (*.f64 t z)) < 9.99999999999999925e-208

   1. Initial program 8.7

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

   2. Simplified 4.9

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 x z) (-.f64 y t))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 z (-.f64 y t))))): 58 points increase in error, 53 points decrease in error
      (*.f64 2 (/.f64 x (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))))): 5 points increase in error, 1 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 x) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x 2)) (-.f64 (*.f64 y z) (*.f64 t z))): 0 points increase in error, 0 points decrease in error

   if 1.00000000000000005e236 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 19.7

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

   2. Simplified 13.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}} \]
      Proof
      (/.f64 x (/.f64 (*.f64 z (-.f64 y t)) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))) 2)): 5 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{y - t}}{-z}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{-207}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{+236}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+267}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+200}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 2
Error	2.3
Cost	840

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{if}\;z \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.4
Cost	840

\[\begin{array}{l} t_1 := \frac{2}{y - t}\\ \mathbf{if}\;z \leq -10:\\ \;\;\;\;\frac{x}{z} \cdot t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternative 5
Error	2.4
Cost	840

\[\begin{array}{l} t_1 := \frac{x \cdot \frac{2}{y - t}}{z}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.6
Cost	712

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 8
Error	18.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 9
Error	18.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 10
Error	31.1
Cost	448

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

Reproduce

herbie shell --seed 2022325 
(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))))