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

Average Error: 10.3 → 0.8

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t_1 \leq -1.794658762741621 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \end{array} \]

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 (+ x (/ (* (- y z) t) (- a z)))))
   (if (<= t_1 (- INFINITY))
     (+ x (* (- y z) (/ t (- a z))))
     (if (<= t_1 -1.794658762741621e-119)
       t_1
       (+ x (* t (- (/ y (- a z)) (/ z (- a z)))))))))

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 = x + (((y - z) * t) / (a - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else if (t_1 <= -1.794658762741621e-119) {
		tmp = t_1;
	} else {
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * t) / (a - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else if (t_1 <= -1.794658762741621e-119) {
		tmp = t_1;
	} else {
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = x + (((y - z) * t) / (a - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x + ((y - z) * (t / (a - z)))
	elif t_1 <= -1.794658762741621e-119:
		tmp = t_1
	else:
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))))
	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(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	elseif (t_1 <= -1.794658762741621e-119)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(t * Float64(Float64(y / Float64(a - z)) - Float64(z / Float64(a - z)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) * t) / (a - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x + ((y - z) * (t / (a - z)));
	elseif (t_1 <= -1.794658762741621e-119)
		tmp = t_1;
	else
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	end
	tmp_2 = 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[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1.794658762741621e-119], t$95$1, N[(x + N[(t * N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.3
Target	0.7
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.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 0.2

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

   3. Applied egg-rr 0.2

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))) < -1.7946587627416209e-119

   1. Initial program 0.1

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

   2. Simplified 2.3

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

   3. Taylor expanded in y around 0 0.1

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

   4. Simplified 2.0

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

   5. Applied egg-rr 0.1

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

   if -1.7946587627416209e-119 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)))

   1. Initial program 9.2

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

   2. Simplified 4.0

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

   3. Taylor expanded in y around 0 9.2

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

   4. Simplified 1.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \leq -1.794658762741621 \cdot 10^{-119}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))