Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Accuracy: 74.8% → 89.5%

Time: 12.9s

Precision: binary64

Cost: 1097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+59} \lor \neg \left(t \leq 6.5 \cdot 10^{+40}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \frac{t - z}{\frac{a - t}{y}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.3e+59) (not (<= t 6.5e+40)))
   (+ x (* (/ y t) (- z a)))
   (+ x (+ y (/ (- t z) (/ (- a t) y))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.3e+59) || !(t <= 6.5e+40)) {
		tmp = x + ((y / t) * (z - a));
	} else {
		tmp = x + (y + ((t - z) / ((a - t) / y)));
	}
	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) * y) / (a - t))
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 ((t <= (-2.3d+59)) .or. (.not. (t <= 6.5d+40))) then
        tmp = x + ((y / t) * (z - a))
    else
        tmp = x + (y + ((t - z) / ((a - t) / y)))
    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) * y) / (a - t));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.3e+59) || !(t <= 6.5e+40)) {
		tmp = x + ((y / t) * (z - a));
	} else {
		tmp = x + (y + ((t - z) / ((a - t) / y)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.3e+59) or not (t <= 6.5e+40):
		tmp = x + ((y / t) * (z - a))
	else:
		tmp = x + (y + ((t - z) / ((a - t) / y)))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.3e+59) || !(t <= 6.5e+40))
		tmp = Float64(x + Float64(Float64(y / t) * Float64(z - a)));
	else
		tmp = Float64(x + Float64(y + Float64(Float64(t - z) / Float64(Float64(a - t) / y))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.3e+59) || ~((t <= 6.5e+40)))
		tmp = x + ((y / t) * (z - a));
	else
		tmp = x + (y + ((t - z) / ((a - t) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.3e+59], N[Not[LessEqual[t, 6.5e+40]], $MachinePrecision]], N[(x + N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y + N[(N[(t - z), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	74.8%
Target	86.9%
Herbie	89.5%

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if t < -2.30000000000000008e59 or 6.5000000000000001e40 < t

   1. Initial program 56.6%

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

   2. Simplified 84.2%

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

      [Start] 56.6	\[ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
      associate--l+ [=>] 62.5	\[ \color{blue}{x + \left(y - \frac{\left(z - t\right) \cdot y}{a - t}\right)} \]
      sub-neg [=>] 62.5	\[ x + \color{blue}{\left(y + \left(-\frac{\left(z - t\right) \cdot y}{a - t}\right)\right)} \]
      +-commutative [=>] 62.5	\[ x + \color{blue}{\left(\left(-\frac{\left(z - t\right) \cdot y}{a - t}\right) + y\right)} \]
      associate-/l* [=>] 79.1	\[ x + \left(\left(-\color{blue}{\frac{z - t}{\frac{a - t}{y}}}\right) + y\right) \]
      distribute-neg-frac [=>] 79.1	\[ x + \left(\color{blue}{\frac{-\left(z - t\right)}{\frac{a - t}{y}}} + y\right) \]
      associate-/r/ [=>] 84.2	\[ x + \left(\color{blue}{\frac{-\left(z - t\right)}{a - t} \cdot y} + y\right) \]
      fma-def [=>] 84.2	\[ x + \color{blue}{\mathsf{fma}\left(\frac{-\left(z - t\right)}{a - t}, y, y\right)} \]
      sub-neg [=>] 84.2	\[ x + \mathsf{fma}\left(\frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a - t}, y, y\right) \]
      +-commutative [=>] 84.2	\[ x + \mathsf{fma}\left(\frac{-\color{blue}{\left(\left(-t\right) + z\right)}}{a - t}, y, y\right) \]
      distribute-neg-in [=>] 84.2	\[ x + \mathsf{fma}\left(\frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a - t}, y, y\right) \]
      unsub-neg [=>] 84.2	\[ x + \mathsf{fma}\left(\frac{\color{blue}{\left(-\left(-t\right)\right) - z}}{a - t}, y, y\right) \]
      remove-double-neg [=>] 84.2	\[ x + \mathsf{fma}\left(\frac{\color{blue}{t} - z}{a - t}, y, y\right) \]

   3. Taylor expanded in t around inf 63.2%

      \[\leadsto x + \color{blue}{\left(y + \left(\frac{y \cdot \left(z - a\right)}{t} + -1 \cdot y\right)\right)} \]

   4. Simplified 85.5%

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

      [Start] 63.2	\[ x + \left(y + \left(\frac{y \cdot \left(z - a\right)}{t} + -1 \cdot y\right)\right) \]
      +-commutative [=>] 63.2	\[ x + \left(y + \color{blue}{\left(-1 \cdot y + \frac{y \cdot \left(z - a\right)}{t}\right)}\right) \]
      associate-+r+ [=>] 73.9	\[ x + \color{blue}{\left(\left(y + -1 \cdot y\right) + \frac{y \cdot \left(z - a\right)}{t}\right)} \]
      distribute-rgt1-in [=>] 73.9	\[ x + \left(\color{blue}{\left(-1 + 1\right) \cdot y} + \frac{y \cdot \left(z - a\right)}{t}\right) \]
      metadata-eval [=>] 73.9	\[ x + \left(\color{blue}{0} \cdot y + \frac{y \cdot \left(z - a\right)}{t}\right) \]
      mul0-lft [=>] 73.9	\[ x + \left(\color{blue}{0} + \frac{y \cdot \left(z - a\right)}{t}\right) \]
      +-lft-identity [=>] 73.9	\[ x + \color{blue}{\frac{y \cdot \left(z - a\right)}{t}} \]
      associate-/l* [=>] 85.4	\[ x + \color{blue}{\frac{y}{\frac{t}{z - a}}} \]
      associate-/r/ [=>] 85.5	\[ x + \color{blue}{\frac{y}{t} \cdot \left(z - a\right)} \]

   if -2.30000000000000008e59 < t < 6.5000000000000001e40

   1. Initial program 88.9%

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

   2. Simplified 92.5%

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

      [Start] 88.9	\[ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
      associate--l+ [=>] 89.9	\[ \color{blue}{x + \left(y - \frac{\left(z - t\right) \cdot y}{a - t}\right)} \]
      associate-/l* [=>] 92.5	\[ x + \left(y - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+59} \lor \neg \left(t \leq 6.5 \cdot 10^{+40}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \frac{t - z}{\frac{a - t}{y}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.4%
Cost	1229

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+66} \lor \neg \left(t \leq 1.7 \cdot 10^{+146}\right) \land t \leq 7.6 \cdot 10^{+218}:\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{t - z}{a - t} + 1\right)\\ \end{array} \]

Alternative 2
Accuracy	75.2%
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;y \cdot \frac{z}{t - a}\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-171}:\\ \;\;\;\;x - y \cdot \frac{a}{t}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Accuracy	75.1%
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a - t}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-172}:\\ \;\;\;\;x - y \cdot \frac{a}{t}\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Accuracy	76.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-49} \lor \neg \left(t \leq 1.4 \cdot 10^{+38}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Accuracy	81.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-49} \lor \neg \left(t \leq 2.1 \cdot 10^{+35}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{z}{\frac{a}{y}}\right)\\ \end{array} \]

Alternative 6
Accuracy	68.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z}{t - a}\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7
Accuracy	76.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+31}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-45}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8
Accuracy	69.2%
Cost	457

\[\begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{-102} \lor \neg \left(a \leq 2.1 \cdot 10^{-43}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	55.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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