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

Average Error: 10.9 → 1.1

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ t_2 := \sqrt[3]{\frac{t}{\frac{a - z}{y}}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot t}{a - z} + \left(x - \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot {t_2}^{2} + \left(t + x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))) (t_2 (cbrt (/ t (/ (- a z) y)))))
   (if (<= t_1 (- INFINITY))
     (fma (- y z) (/ t (- a z)) x)
     (if (<= t_1 2e+294)
       (+ (/ (* y t) (- a z)) (- x (/ (* z t) (- a z))))
       (+ (* t_2 (pow t_2 2.0)) (+ t x))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double t_2 = cbrt((t / ((a - z) / y)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (t_1 <= 2e+294) {
		tmp = ((y * t) / (a - z)) + (x - ((z * t) / (a - z)));
	} else {
		tmp = (t_2 * pow(t_2, 2.0)) + (t + x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	t_2 = cbrt(Float64(t / Float64(Float64(a - z) / y)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (t_1 <= 2e+294)
		tmp = Float64(Float64(Float64(y * t) / Float64(a - z)) + Float64(x - Float64(Float64(z * t) / Float64(a - z))));
	else
		tmp = Float64(Float64(t_2 * (t_2 ^ 2.0)) + Float64(t + x));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+294], N[(N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(z * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t + x), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	10.9
Target	0.6
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.1

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 2.00000000000000013e294

   1. Initial program 0.2

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

   2. Simplified 3.7

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

   3. Taylor expanded in y around 0 0.2

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

   if 2.00000000000000013e294 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 61.9

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

   2. Simplified 1.0

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

   3. Taylor expanded in y around 0 61.9

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

   4. Applied egg-rr 48.2

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{t}{\frac{a - z}{y}}}\right)}^{2} \cdot \sqrt[3]{\frac{t}{\frac{a - z}{y}}}} + \left(x + -1 \cdot \frac{t \cdot z}{a - z}\right) \]

   5. Taylor expanded in z around inf 10.6

      \[\leadsto {\left(\sqrt[3]{\frac{t}{\frac{a - z}{y}}}\right)}^{2} \cdot \sqrt[3]{\frac{t}{\frac{a - z}{y}}} + \left(x + -1 \cdot \color{blue}{\left(-1 \cdot t\right)}\right) \]

   6. Simplified 10.6

      \[\leadsto {\left(\sqrt[3]{\frac{t}{\frac{a - z}{y}}}\right)}^{2} \cdot \sqrt[3]{\frac{t}{\frac{a - z}{y}}} + \left(x + -1 \cdot \color{blue}{\left(-t\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot t}{a - z} + \left(x - \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{t}{\frac{a - z}{y}}} \cdot {\left(\sqrt[3]{\frac{t}{\frac{a - z}{y}}}\right)}^{2} + \left(t + x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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