Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.9 → 5.9

Time: 5.3s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (/ (+ x x) (* 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) {
	return (x + x) / (z * (y - t));
}

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
    code = (x + x) / (z * (y - t))
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) {
	return (x + x) / (z * (y - t));
}

Python

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

↓

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

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)
	return Float64(Float64(x + x) / Float64(z * Float64(y - t)))
end

MATLAB

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

↓

function tmp = code(x, y, z, t)
	tmp = (x + x) / (z * (y - t));
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_] := N[(N[(x + x), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.9
Target	2.5
Herbie	5.9

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

Derivation

1. Initial program 6.9

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

2. Simplified 5.9

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

   [Start] 6.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   metadata-eval [<=] 6.9	\[ \frac{x \cdot \color{blue}{\left(1 + 1\right)}}{y \cdot z - t \cdot z} \]
   metadata-eval [<=] 6.9	\[ \frac{x \cdot \left(\color{blue}{\frac{2}{2}} + 1\right)}{y \cdot z - t \cdot z} \]
   metadata-eval [<=] 6.9	\[ \frac{x \cdot \left(\frac{2}{2} + \color{blue}{\frac{2}{2}}\right)}{y \cdot z - t \cdot z} \]
   rational_best_oopsla_all_46_json_45_simplify-23 [<=] 6.9	\[ \frac{\color{blue}{\frac{2}{2} \cdot x + x \cdot \frac{2}{2}}}{y \cdot z - t \cdot z} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 6.9	\[ \frac{\color{blue}{x \cdot \frac{2}{2}} + x \cdot \frac{2}{2}}{y \cdot z - t \cdot z} \]
   metadata-eval [=>] 6.9	\[ \frac{x \cdot \color{blue}{1} + x \cdot \frac{2}{2}}{y \cdot z - t \cdot z} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 6.9	\[ \frac{\color{blue}{x} + x \cdot \frac{2}{2}}{y \cdot z - t \cdot z} \]
   metadata-eval [=>] 6.9	\[ \frac{x + x \cdot \color{blue}{1}}{y \cdot z - t \cdot z} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 6.9	\[ \frac{x + \color{blue}{x}}{y \cdot z - t \cdot z} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 6.9	\[ \frac{x + x}{y \cdot z - \color{blue}{z \cdot t}} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 5.9	\[ \frac{x + x}{\color{blue}{z \cdot \left(y - t\right)}} \]

3. Final simplification 5.9

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

Alternatives

Alternative 1
Error	17.0
Cost	712

\[\begin{array}{l} t_1 := -2 \cdot \frac{x}{t \cdot z}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x + x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	31.2
Cost	448

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

Reproduce

herbie shell --seed 2023090 
(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))))