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

Average Error: 10.9 → 0.3

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ t_2 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{y \cdot t}{a - z} + \left(x - \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma (/ (- y z) (- a z)) t x)) (t_2 (/ (* (- y z) t) (- a z))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 5e+295)
       (+ (/ (* y t) (- a z)) (- x (/ (* z t) (- a z))))
       t_1))))

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 = fma(((y - z) / (a - z)), t, x);
	double t_2 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+295) {
		tmp = ((y * t) / (a - z)) + (x - ((z * t) / (a - z)));
	} else {
		tmp = t_1;
	}
	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 = fma(Float64(Float64(y - z) / Float64(a - z)), t, x)
	t_2 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+295)
		tmp = Float64(Float64(Float64(y * t) / Float64(a - z)) + Float64(x - Float64(Float64(z * t) / Float64(a - z))));
	else
		tmp = t_1;
	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] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+295], 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], t$95$1]]]]

Target

Original	10.9
Target	0.6
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0 or 4.99999999999999991e295 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 63.3

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.4

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{{\left(\frac{a - z}{t}\right)}^{-1}}, x\right) \]

   4. Applied egg-rr 0.3

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

   5. Applied egg-rr 0.4

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 4.99999999999999991e295

   1. Initial program 0.3

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

   2. Simplified 3.3

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

   3. Taylor expanded in y around 0 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{y \cdot t}{a - z} + \left(x - \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022182 
(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))))