Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 10.6 → 1.3

Time: 11.2s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{y}{\frac{z - a}{z - t}} + x \]

FPCore

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

↓

(FPCore (x y z t a) :precision binary64 (+ (/ y (/ (- z a) (- z t))) x))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	return (y / ((z - a) / (z - t))) + x;
}

Fortran

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

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (y / ((z - a) / (z - t))) + x
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

function code(x, y, z, t, a)
	return Float64(Float64(y / Float64(Float64(z - a) / Float64(z - t))) + x)
end

MATLAB

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

↓

function tmp = code(x, y, z, t, a)
	tmp = (y / ((z - a) / (z - t))) + x;
end

Wolfram

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

↓

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

Target

Original	10.6
Target	1.3
Herbie	1.3

\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

1. Initial program 10.6

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

2. Simplified 1.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \]
   Proof
   (fma.f64 y (/.f64 (-.f64 z t) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) x)): 1 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) x): 42 points increase in error, 19 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 1.3

   \[\leadsto \color{blue}{\frac{y}{\frac{z - a}{z - t}} + x} \]

4. Final simplification 1.3

   \[\leadsto \frac{y}{\frac{z - a}{z - t}} + x \]

Alternatives

Alternative 1
Error	13.5
Cost	1236

\[\begin{array}{l} \mathbf{if}\;a \leq -2.9713530403000132 \cdot 10^{+85}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -3.219066739365809 \cdot 10^{-51}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;a \leq -1.6855840506418524 \cdot 10^{-127}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq -2.6646121191957703 \cdot 10^{-167}:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}}\\ \mathbf{elif}\;a \leq 3211383327.1486025:\\ \;\;\;\;x + \left(y - y \cdot \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	908

\[\begin{array}{l} \mathbf{if}\;z \leq -3.205958724596176 \cdot 10^{-62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 4.813118325158483 \cdot 10^{-18}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.4372556727175404 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{y}{\frac{-z}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 3
Error	14.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -5.739658963249961 \cdot 10^{+109}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 3211383327.1486025:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 4
Error	12.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1020398381747617 \cdot 10^{-17}:\\ \;\;\;\;x + z \cdot \frac{y}{z - a}\\ \mathbf{elif}\;a \leq 3211383327.1486025:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 5
Error	12.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1020398381747617 \cdot 10^{-17}:\\ \;\;\;\;x + z \cdot \frac{y}{z - a}\\ \mathbf{elif}\;a \leq 3211383327.1486025:\\ \;\;\;\;x + \left(y - y \cdot \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 6
Error	14.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.0038071639322261 \cdot 10^{-63}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 8.492819633006746 \cdot 10^{+26}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	14.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.205958724596176 \cdot 10^{-62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 8.492819633006746 \cdot 10^{+26}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	20.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5.739658963249961 \cdot 10^{+109}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.732455245354841 \cdot 10^{+37}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	26.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -6.197510220527438 \cdot 10^{-214}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7781682491492695 \cdot 10^{-182}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	28.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))