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

Average Error: 1.3 → 0.8

Time: 15.9s

Precision: binary64

Cost: 1096

Math

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

↓

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{\frac{z - t}{\frac{1}{y}}}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- z a))))))
   (if (<= y -8.8e+46)
     t_1
     (if (<= y 1.2e-161) (+ x (/ (/ (- z t) (/ 1.0 y)) (- z a))) t_1))))

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 t_1 = x + (y * ((z - t) / (z - a)));
	double tmp;
	if (y <= -8.8e+46) {
		tmp = t_1;
	} else if (y <= 1.2e-161) {
		tmp = x + (((z - t) / (1.0 / y)) / (z - a));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (z - a)))
    if (y <= (-8.8d+46)) then
        tmp = t_1
    else if (y <= 1.2d-161) then
        tmp = x + (((z - t) / (1.0d0 / y)) / (z - a))
    else
        tmp = t_1
    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 t_1 = x + (y * ((z - t) / (z - a)));
	double tmp;
	if (y <= -8.8e+46) {
		tmp = t_1;
	} else if (y <= 1.2e-161) {
		tmp = x + (((z - t) / (1.0 / y)) / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (z - a)))
	tmp = 0
	if y <= -8.8e+46:
		tmp = t_1
	elif y <= 1.2e-161:
		tmp = x + (((z - t) / (1.0 / y)) / (z - a))
	else:
		tmp = t_1
	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)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
	tmp = 0.0
	if (y <= -8.8e+46)
		tmp = t_1;
	elseif (y <= 1.2e-161)
		tmp = Float64(x + Float64(Float64(Float64(z - t) / Float64(1.0 / y)) / Float64(z - a)));
	else
		tmp = t_1;
	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)
	t_1 = x + (y * ((z - t) / (z - a)));
	tmp = 0.0;
	if (y <= -8.8e+46)
		tmp = t_1;
	elseif (y <= 1.2e-161)
		tmp = x + (((z - t) / (1.0 / y)) / (z - a));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e+46], t$95$1, If[LessEqual[y, 1.2e-161], N[(x + N[(N[(N[(z - t), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	1.3
Target	1.2
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000001e46 or 1.19999999999999999e-161 < y

   1. Initial program 0.8

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

   if -8.8000000000000001e46 < y < 1.19999999999999999e-161

   1. Initial program 2.1

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

   2. Simplified 2.9

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

      [Start] 2.1	\[ x + y \cdot \frac{z - t}{z - a} \]
      rational.json-simplify-50 [=>] 2.1	\[ x + y \cdot \color{blue}{\frac{-\left(z - t\right)}{a - z}} \]
      rational.json-simplify-8 [=>] 2.1	\[ x + y \cdot \frac{\color{blue}{\left(z - t\right) \cdot -1}}{a - z} \]
      rational.json-simplify-49 [=>] 2.1	\[ x + y \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - z}\right)} \]
      rational.json-simplify-43 [<=] 2.1	\[ x + \color{blue}{\frac{z - t}{a - z} \cdot \left(y \cdot -1\right)} \]
      rational.json-simplify-2 [=>] 2.1	\[ x + \color{blue}{\left(y \cdot -1\right) \cdot \frac{z - t}{a - z}} \]
      rational.json-simplify-50 [=>] 2.1	\[ x + \left(y \cdot -1\right) \cdot \color{blue}{\frac{-\left(z - t\right)}{z - a}} \]
      rational.json-simplify-8 [=>] 2.1	\[ x + \left(y \cdot -1\right) \cdot \frac{\color{blue}{\left(z - t\right) \cdot -1}}{z - a} \]
      rational.json-simplify-2 [=>] 2.1	\[ x + \left(y \cdot -1\right) \cdot \frac{\color{blue}{-1 \cdot \left(z - t\right)}}{z - a} \]
      rational.json-simplify-49 [=>] 2.1	\[ x + \left(y \cdot -1\right) \cdot \color{blue}{\left(\left(z - t\right) \cdot \frac{-1}{z - a}\right)} \]
      rational.json-simplify-43 [=>] 2.9	\[ x + \color{blue}{\left(z - t\right) \cdot \left(\frac{-1}{z - a} \cdot \left(y \cdot -1\right)\right)} \]
      rational.json-simplify-9 [=>] 2.9	\[ x + \left(z - t\right) \cdot \left(\frac{-1}{z - a} \cdot \color{blue}{\left(-y\right)}\right) \]
      rational.json-simplify-10 [=>] 2.9	\[ x + \left(z - t\right) \cdot \left(\frac{-1}{z - a} \cdot \color{blue}{\frac{y}{-1}}\right) \]
      rational.json-simplify-55 [=>] 2.9	\[ x + \left(z - t\right) \cdot \color{blue}{\frac{\frac{y}{-1}}{\frac{z - a}{-1}}} \]
      rational.json-simplify-10 [<=] 2.9	\[ x + \left(z - t\right) \cdot \frac{\color{blue}{-y}}{\frac{z - a}{-1}} \]
      rational.json-simplify-9 [<=] 2.9	\[ x + \left(z - t\right) \cdot \frac{\color{blue}{y \cdot -1}}{\frac{z - a}{-1}} \]
      rational.json-simplify-10 [<=] 2.9	\[ x + \left(z - t\right) \cdot \frac{y \cdot -1}{\color{blue}{-\left(z - a\right)}} \]
      rational.json-simplify-13 [<=] 2.9	\[ x + \left(z - t\right) \cdot \frac{y \cdot -1}{\color{blue}{0 - \left(z - a\right)}} \]
      rational.json-simplify-45 [<=] 2.9	\[ x + \left(z - t\right) \cdot \frac{y \cdot -1}{\color{blue}{a - \left(z - 0\right)}} \]
      rational.json-simplify-5 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{y \cdot -1}{a - \color{blue}{z}} \]
      rational.json-simplify-50 [=>] 2.9	\[ x + \left(z - t\right) \cdot \color{blue}{\frac{-y \cdot -1}{z - a}} \]
      rational.json-simplify-12 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{\color{blue}{0 - y \cdot -1}}{z - a} \]
      rational.json-simplify-9 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{0 - \color{blue}{\left(-y\right)}}{z - a} \]
      rational.json-simplify-12 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{0 - \color{blue}{\left(0 - y\right)}}{z - a} \]
      rational.json-simplify-45 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{\color{blue}{y - \left(0 - 0\right)}}{z - a} \]
      metadata-eval [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{y - \color{blue}{0}}{z - a} \]
      rational.json-simplify-5 [=>] 2.9	\[ x + \left(z - t\right) \cdot \frac{\color{blue}{y}}{z - a} \]

   3. Applied egg-rr 0.8

      \[\leadsto x + \color{blue}{\frac{\frac{z - t}{\frac{1}{y}}}{z - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+46}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{\frac{z - t}{\frac{1}{y}}}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	1220

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+249}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot t_1\\ \end{array} \]

Alternative 2
Error	0.8
Cost	1220

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+249}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a - z}{y}}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot t_1\\ \end{array} \]

Alternative 3
Error	1.3
Cost	968

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-289}:\\ \;\;\;\;x + \frac{z - t}{\frac{z - a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.5
Cost	968

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-17}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -300000000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-79}:\\ \;\;\;\;y \cdot \frac{t}{-z} + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+34}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Error	10.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+97}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	10.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+45}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	8.4
Cost	840

\[\begin{array}{l} t_1 := y \cdot \frac{z}{z - a} + x\\ \mathbf{if}\;z \leq -6 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	20.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 11
Error	26.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-234}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-136}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	29.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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)))))