Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.3 → 2.2

Time: 5.3s

Precision: binary64

Cost: 1100

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

↓

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-182}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y z))) (t_2 (/ (* y x) z)))
   (if (<= (/ y z) (- INFINITY))
     t_2
     (if (<= (/ y z) -2e-149) t_1 (if (<= (/ y z) 2e-182) t_2 t_1)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -((double) INFINITY)) {
		tmp = t_2;
	} else if ((y / z) <= -2e-149) {
		tmp = t_1;
	} else if ((y / z) <= 2e-182) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if ((y / z) <= -2e-149) {
		tmp = t_1;
	} else if ((y / z) <= 2e-182) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x * (y / z)
	t_2 = (y * x) / z
	tmp = 0
	if (y / z) <= -math.inf:
		tmp = t_2
	elif (y / z) <= -2e-149:
		tmp = t_1
	elif (y / z) <= 2e-182:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / z))
	t_2 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= Float64(-Inf))
		tmp = t_2;
	elseif (Float64(y / z) <= -2e-149)
		tmp = t_1;
	elseif (Float64(y / z) <= 2e-182)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / z);
	t_2 = (y * x) / z;
	tmp = 0.0;
	if ((y / z) <= -Inf)
		tmp = t_2;
	elseif ((y / z) <= -2e-149)
		tmp = t_1;
	elseif ((y / z) <= 2e-182)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], (-Infinity)], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -2e-149], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 2e-182], t$95$2, t$95$1]]]]]

Target

Original	14.3
Target	1.6
Herbie	2.2

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < -inf.0 or -1.99999999999999996e-149 < (/.f64 y z) < 2.0000000000000001e-182

   1. Initial program 19.1

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Taylor expanded in x around 0 1.2

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

   if -inf.0 < (/.f64 y z) < -1.99999999999999996e-149 or 2.0000000000000001e-182 < (/.f64 y z)

   1. Initial program 11.8

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Taylor expanded in y around 0 2.7

      \[\leadsto x \cdot \color{blue}{\frac{y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-182}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	320

\[x \cdot \frac{y}{z} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))