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

Average Error: 2.38% → 0.83%

Time: 12.5s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-27} \lor \neg \left(y \leq 0.005\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -5.8e-27) (not (<= y 0.005)))
   (+ x (* y (/ (- z t) (- z a))))
   (+ x (/ (* y (- z t)) (- z a)))))

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) {
	double tmp;
	if ((y <= -5.8e-27) || !(y <= 0.005)) {
		tmp = x + (y * ((z - t) / (z - a)));
	} else {
		tmp = x + ((y * (z - t)) / (z - a));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y <= (-5.8d-27)) .or. (.not. (y <= 0.005d0))) then
        tmp = x + (y * ((z - t) / (z - a)))
    else
        tmp = x + ((y * (z - t)) / (z - a))
    end if
    code = tmp
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) {
	double tmp;
	if ((y <= -5.8e-27) || !(y <= 0.005)) {
		tmp = x + (y * ((z - t) / (z - a)));
	} else {
		tmp = x + ((y * (z - t)) / (z - a));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -5.8e-27) or not (y <= 0.005):
		tmp = x + (y * ((z - t) / (z - a)))
	else:
		tmp = x + ((y * (z - t)) / (z - a))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -5.8e-27) || !(y <= 0.005))
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))));
	else
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -5.8e-27) || ~((y <= 0.005)))
		tmp = x + (y * ((z - t) / (z - a)));
	else
		tmp = x + ((y * (z - t)) / (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -5.8e-27], N[Not[LessEqual[y, 0.005]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.38%
Target	2.13%
Herbie	0.83%

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

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000008e-27 or 0.0050000000000000001 < y

   1. Initial program 1.1

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

   if -5.80000000000000008e-27 < y < 0.0050000000000000001

   1. Initial program 3.62

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

   2. Simplified 0.58

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

      [Start] 3.62	\[ x + y \cdot \frac{z - t}{z - a} \]
      associate-*r/ [=>] 0.58	\[ x + \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.83

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-27} \lor \neg \left(y \leq 0.005\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.55%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+27}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \end{array} \]

Alternative 2
Error	23.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -2850000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 29:\\ \;\;\;\;x - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq 1000000000000:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	3.58%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+27}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-304}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{z - a}{y}}\\ \end{array} \]

Alternative 4
Error	33.49%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -440000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-266}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 5000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	18.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -820000 \lor \neg \left(z \leq 3.4 \cdot 10^{-70}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	15.98%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-113} \lor \neg \left(z \leq 8.2 \cdot 10^{-45}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Error	14.92%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-48} \lor \neg \left(z \leq 3.2 \cdot 10^{-42}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \end{array} \]

Alternative 8
Error	22.53%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1200000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 50000000000:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9
Error	22.48%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -460000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1600000000000:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	22.43%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1040000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 190000000000:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Error	22.39%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	2.38%
Cost	704

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

Alternative 13
Error	31.47%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -112000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1150000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 14
Error	42.97%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-232}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	45.46%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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