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

Percentage Accurate: 98.2% → 98.8%

Time: 14.3s

Precision: binary64

Cost: 1604

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+184}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{\frac{a - t}{z}} - \frac{y}{\frac{a}{t} + -1}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) -4e+184)
   (+ (* (- t z) (/ y (- t a))) x)
   (+ x (- (/ y (/ (- a t) z)) (/ y (+ (/ a t) -1.0))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= -4e+184) {
		tmp = ((t - z) * (y / (t - a))) + x;
	} else {
		tmp = x + ((y / ((a - t) / z)) - (y / ((a / t) + -1.0)));
	}
	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) / (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 (((z - t) / (a - t)) <= (-4d+184)) then
        tmp = ((t - z) * (y / (t - a))) + x
    else
        tmp = x + ((y / ((a - t) / z)) - (y / ((a / t) + (-1.0d0))))
    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) / (a - t)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= -4e+184) {
		tmp = ((t - z) * (y / (t - a))) + x;
	} else {
		tmp = x + ((y / ((a - t) / z)) - (y / ((a / t) + -1.0)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (a - t)) <= -4e+184:
		tmp = ((t - z) * (y / (t - a))) + x
	else:
		tmp = x + ((y / ((a - t) / z)) - (y / ((a / t) + -1.0)))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= -4e+184)
		tmp = Float64(Float64(Float64(t - z) * Float64(y / Float64(t - a))) + x);
	else
		tmp = Float64(x + Float64(Float64(y / Float64(Float64(a - t) / z)) - Float64(y / Float64(Float64(a / t) + -1.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (a - t)) <= -4e+184)
		tmp = ((t - z) * (y / (t - a))) + x;
	else
		tmp = x + ((y / ((a - t) / z)) - (y / ((a / t) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], -4e+184], N[(N[(N[(t - z), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y / N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[(a / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	98.2%
Target	99.3%
Herbie	98.8%

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -4.00000000000000007e184

   1. Initial program 79.4%

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

   2. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - z, \frac{y}{t - a}, x\right)} \]
      Step-by-step derivation

      [Start] 79.4%	\[ x + y \cdot \frac{z - t}{a - t} \]
      +-commutative [=>] 79.4%	\[ \color{blue}{y \cdot \frac{z - t}{a - t} + x} \]
      *-commutative [=>] 79.4%	\[ \color{blue}{\frac{z - t}{a - t} \cdot y} + x \]
      associate-*l/ [=>] 99.8%	\[ \color{blue}{\frac{\left(z - t\right) \cdot y}{a - t}} + x \]
      sub-neg [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{a + \left(-t\right)}} + x \]
      +-commutative [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(-t\right) + a}} + x \]
      neg-sub0 [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]
      associate-+l- [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]
      sub0-neg [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{-\left(t - a\right)}} + x \]
      neg-mul-1 [=>] 99.8%	\[ \frac{\left(z - t\right) \cdot y}{\color{blue}{-1 \cdot \left(t - a\right)}} + x \]
      times-frac [=>] 99.9%	\[ \color{blue}{\frac{z - t}{-1} \cdot \frac{y}{t - a}} + x \]
      fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(\frac{z - t}{-1}, \frac{y}{t - a}, x\right)} \]
      sub-neg [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{z + \left(-t\right)}}{-1}, \frac{y}{t - a}, x\right) \]
      +-commutative [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(-t\right) + z}}{-1}, \frac{y}{t - a}, x\right) \]
      neg-sub0 [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]
      associate-+l- [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]
      sub0-neg [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]
      neg-mul-1 [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]
      *-commutative [=>] 99.9%	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(t - z\right) \cdot -1}}{-1}, \frac{y}{t - a}, x\right) \]
      associate-/l* [=>] 99.9%	\[ \mathsf{fma}\left(\color{blue}{\frac{t - z}{\frac{-1}{-1}}}, \frac{y}{t - a}, x\right) \]
      metadata-eval [=>] 99.9%	\[ \mathsf{fma}\left(\frac{t - z}{\color{blue}{1}}, \frac{y}{t - a}, x\right) \]
      /-rgt-identity [=>] 99.9%	\[ \mathsf{fma}\left(\color{blue}{t - z}, \frac{y}{t - a}, x\right) \]

   3. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \mathsf{fma}\left(t - z, \frac{y}{t - a}, x\right) \]
      fma-udef [=>] 99.9%	\[ \color{blue}{\left(t - z\right) \cdot \frac{y}{t - a} + x} \]

   if -4.00000000000000007e184 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.6%

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

   2. Taylor expanded in z around 0 82.4%

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

   3. Simplified 98.7%

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

      [Start] 82.4%	\[ x + \left(\frac{y \cdot z}{a - t} + -1 \cdot \frac{y \cdot t}{a - t}\right) \]
      mul-1-neg [=>] 82.4%	\[ x + \left(\frac{y \cdot z}{a - t} + \color{blue}{\left(-\frac{y \cdot t}{a - t}\right)}\right) \]
      unsub-neg [=>] 82.4%	\[ x + \color{blue}{\left(\frac{y \cdot z}{a - t} - \frac{y \cdot t}{a - t}\right)} \]
      associate-/l* [=>] 85.0%	\[ x + \left(\color{blue}{\frac{y}{\frac{a - t}{z}}} - \frac{y \cdot t}{a - t}\right) \]
      associate-/l* [=>] 98.7%	\[ x + \left(\frac{y}{\frac{a - t}{z}} - \color{blue}{\frac{y}{\frac{a - t}{t}}}\right) \]
      div-sub [=>] 98.7%	\[ x + \left(\frac{y}{\frac{a - t}{z}} - \frac{y}{\color{blue}{\frac{a}{t} - \frac{t}{t}}}\right) \]
      *-inverses [=>] 98.7%	\[ x + \left(\frac{y}{\frac{a - t}{z}} - \frac{y}{\frac{a}{t} - \color{blue}{1}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+184}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{\frac{a - t}{z}} - \frac{y}{\frac{a}{t} + -1}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+184}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{\frac{a - t}{z}} - \frac{y}{\frac{a}{t} + -1}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.7%
Cost	1220

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

Alternative 3
Accuracy	98.8%
Cost	1220

\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+175}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot y\\ \end{array} \]

Alternative 4
Accuracy	82.8%
Cost	972

\[\begin{array}{l} t_1 := x + \left(y - \frac{y}{\frac{t}{z}}\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-66}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{t}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	82.8%
Cost	972

\[\begin{array}{l} t_1 := x + \left(y - \frac{y}{\frac{t}{z}}\right)\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-72}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{t}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-18}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t - z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	77.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-66} \lor \neg \left(t \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 7
Accuracy	82.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-76} \lor \neg \left(t \leq 1.8 \cdot 10^{-10}\right):\\ \;\;\;\;x + \left(y - \frac{y}{\frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 8
Accuracy	62.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+112} \lor \neg \left(y \leq 8 \cdot 10^{+147}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Accuracy	75.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-66} \lor \neg \left(t \leq 7.8 \cdot 10^{-16}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 10
Accuracy	75.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-66} \lor \neg \left(t \leq 1.4 \cdot 10^{-10}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 11
Accuracy	60.6%
Cost	516

\[\begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+239}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \end{array} \]

Alternative 12
Accuracy	60.2%
Cost	192

\[y + x \]

Alternative 13
Accuracy	51.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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