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

Average Error: 16.1 → 6.5

Time: 13.5s

Precision: binary64

Cost: 1160

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+206}:\\ \;\;\;\;x + y \cdot \left(-\frac{a - z}{t}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+76}:\\ \;\;\;\;x + y \cdot \left(1 + \left(-\frac{z - t}{a - t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-\frac{z}{a - t}\right)\\ \end{array} \]

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 (<= t -6.6e+206)
   (+ x (* y (- (/ (- a z) t))))
   (if (<= t 3.4e+76)
     (+ x (* y (+ 1.0 (- (/ (- z t) (- a t))))))
     (+ x (* y (- (/ z (- a t))))))))

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 <= -6.6e+206) {
		tmp = x + (y * -((a - z) / t));
	} else if (t <= 3.4e+76) {
		tmp = x + (y * (1.0 + -((z - t) / (a - t))));
	} else {
		tmp = x + (y * -(z / (a - t)));
	}
	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 <= (-6.6d+206)) then
        tmp = x + (y * -((a - z) / t))
    else if (t <= 3.4d+76) then
        tmp = x + (y * (1.0d0 + -((z - t) / (a - t))))
    else
        tmp = x + (y * -(z / (a - t)))
    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 <= -6.6e+206) {
		tmp = x + (y * -((a - z) / t));
	} else if (t <= 3.4e+76) {
		tmp = x + (y * (1.0 + -((z - t) / (a - t))));
	} else {
		tmp = x + (y * -(z / (a - t)));
	}
	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 <= -6.6e+206:
		tmp = x + (y * -((a - z) / t))
	elif t <= 3.4e+76:
		tmp = x + (y * (1.0 + -((z - t) / (a - t))))
	else:
		tmp = x + (y * -(z / (a - t)))
	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 <= -6.6e+206)
		tmp = Float64(x + Float64(y * Float64(-Float64(Float64(a - z) / t))));
	elseif (t <= 3.4e+76)
		tmp = Float64(x + Float64(y * Float64(1.0 + Float64(-Float64(Float64(z - t) / Float64(a - t))))));
	else
		tmp = Float64(x + Float64(y * Float64(-Float64(z / Float64(a - t)))));
	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 <= -6.6e+206)
		tmp = x + (y * -((a - z) / t));
	elseif (t <= 3.4e+76)
		tmp = x + (y * (1.0 + -((z - t) / (a - t))));
	else
		tmp = x + (y * -(z / (a - t)));
	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[LessEqual[t, -6.6e+206], N[(x + N[(y * (-N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+76], N[(x + N[(y * N[(1.0 + (-N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * (-N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Target

Original	16.1
Target	8.4
Herbie	6.5

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

Derivation

1. Split input into 3 regimes

2. if t < -6.59999999999999969e206

   1. Initial program 34.8

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

   2. Taylor expanded in y around -inf 11.9

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

   3. Simplified 11.9

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

      [Start] 11.9	\[ y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right) + x \]
      rational_best.json-simplify-1 [=>] 11.9	\[ \color{blue}{x + y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
      rational_best.json-simplify-2 [=>] 11.9	\[ x + y \cdot \left(1 + \color{blue}{\frac{z - t}{a - t} \cdot -1}\right) \]
      rational_best.json-simplify-12 [=>] 11.9	\[ x + y \cdot \left(1 + \color{blue}{\left(-\frac{z - t}{a - t}\right)}\right) \]

   4. Taylor expanded in t around -inf 3.7

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

   5. Simplified 3.7

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

      [Start] 3.7	\[ x + y \cdot \left(-1 \cdot \frac{a - z}{t}\right) \]
      rational_best.json-simplify-2 [=>] 3.7	\[ x + y \cdot \color{blue}{\left(\frac{a - z}{t} \cdot -1\right)} \]
      rational_best.json-simplify-12 [=>] 3.7	\[ x + y \cdot \color{blue}{\left(-\frac{a - z}{t}\right)} \]

   if -6.59999999999999969e206 < t < 3.3999999999999997e76

   1. Initial program 10.2

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

   2. Taylor expanded in y around -inf 5.6

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

   3. Simplified 5.6

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

      [Start] 5.6	\[ y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right) + x \]
      rational_best.json-simplify-1 [=>] 5.6	\[ \color{blue}{x + y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
      rational_best.json-simplify-2 [=>] 5.6	\[ x + y \cdot \left(1 + \color{blue}{\frac{z - t}{a - t} \cdot -1}\right) \]
      rational_best.json-simplify-12 [=>] 5.6	\[ x + y \cdot \left(1 + \color{blue}{\left(-\frac{z - t}{a - t}\right)}\right) \]

   if 3.3999999999999997e76 < t

   1. Initial program 28.4

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

   2. Taylor expanded in y around -inf 11.2

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

   3. Simplified 11.2

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

      [Start] 11.2	\[ y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right) + x \]
      rational_best.json-simplify-1 [=>] 11.2	\[ \color{blue}{x + y \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
      rational_best.json-simplify-2 [=>] 11.2	\[ x + y \cdot \left(1 + \color{blue}{\frac{z - t}{a - t} \cdot -1}\right) \]
      rational_best.json-simplify-12 [=>] 11.2	\[ x + y \cdot \left(1 + \color{blue}{\left(-\frac{z - t}{a - t}\right)}\right) \]

   4. Taylor expanded in z around inf 10.9

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

   5. Simplified 10.9

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

      [Start] 10.9	\[ x + y \cdot \left(-1 \cdot \frac{z}{a - t}\right) \]
      rational_best.json-simplify-2 [=>] 10.9	\[ x + y \cdot \color{blue}{\left(\frac{z}{a - t} \cdot -1\right)} \]
      rational_best.json-simplify-12 [=>] 10.9	\[ x + y \cdot \color{blue}{\left(-\frac{z}{a - t}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+206}:\\ \;\;\;\;x + y \cdot \left(-\frac{a - z}{t}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+76}:\\ \;\;\;\;x + y \cdot \left(1 + \left(-\frac{z - t}{a - t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-\frac{z}{a - t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.6
Cost	4432

\[\begin{array}{l} t_1 := x + y \cdot \left(-\frac{a - z}{t}\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-232}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-\frac{z}{a - t}\right)\\ \end{array} \]

Alternative 2
Error	14.0
Cost	904

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-49}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 0.125:\\ \;\;\;\;\left(1 - \frac{z}{a}\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(-\frac{y \cdot \left(a - z\right)}{t}\right)\\ \end{array} \]

Alternative 3
Error	8.4
Cost	904

\[\begin{array}{l} t_1 := \left(1 - \frac{z}{a}\right) \cdot y + x\\ \mathbf{if}\;a \leq -6.4 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+30}:\\ \;\;\;\;x + y \cdot \left(-\frac{z}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	11.8
Cost	904

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;x + y \cdot \left(-\frac{a - z}{t}\right)\\ \mathbf{elif}\;t \leq 0.0055:\\ \;\;\;\;\left(1 - \frac{z}{a}\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-\frac{z}{a - t}\right)\\ \end{array} \]

Alternative 5
Error	13.1
Cost	840

\[\begin{array}{l} t_1 := x + y \cdot \frac{z}{t}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.42:\\ \;\;\;\;\left(1 - \frac{z}{a}\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.0
Cost	712

\[\begin{array}{l} t_1 := x + y \cdot \frac{z}{t}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+23}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{+206}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	27.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+95}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9
Error	28.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))