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

Average Error: 11.1 → 0.6

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t}} + x\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \end{array} \]

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))))
   (if (<= t_1 -1e+76)
     (+ (/ (- y z) (/ (- a z) t)) x)
     (if (<= t_1 5e+307) (+ t_1 x) (fma (- y z) (/ t (- a z)) 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 tmp;
	if (t_1 <= -1e+76) {
		tmp = ((y - z) / ((a - z) / t)) + x;
	} else if (t_1 <= 5e+307) {
		tmp = t_1 + x;
	} else {
		tmp = fma((y - z), (t / (a - z)), 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))
	tmp = 0.0
	if (t_1 <= -1e+76)
		tmp = Float64(Float64(Float64(y - z) / Float64(Float64(a - z) / t)) + x);
	elseif (t_1 <= 5e+307)
		tmp = Float64(t_1 + x);
	else
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), 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]}, If[LessEqual[t$95$1, -1e+76], N[(N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], N[(t$95$1 + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	11.1
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -1e76

   1. Initial program 31.6

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

   2. Simplified 3.0

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

   3. Applied egg-rr 3.1

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

   4. Applied egg-rr 2.7

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

   if -1e76 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 5e307

   1. Initial program 0.2

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

   if 5e307 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 64.0

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t}} + x\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))