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

Average Error: 14.8 → 0.5

Time: 4.0s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -4.793136556035763 \cdot 10^{+249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -7.851769693067612 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 3.176090190266049 \cdot 10^{-124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 4.2418494777197253 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (/ z y))) (t_2 (* y (/ x z))))
   (if (<= (/ y z) -4.793136556035763e+249)
     t_2
     (if (<= (/ y z) -7.851769693067612e-293)
       t_1
       (if (<= (/ y z) 3.176090190266049e-124)
         t_2
         (if (<= (/ y z) 4.2418494777197253e+219)
           t_1
           (* (/ 1.0 z) (* y x))))))))

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 / (z / y);
	double t_2 = y * (x / z);
	double tmp;
	if ((y / z) <= -4.793136556035763e+249) {
		tmp = t_2;
	} else if ((y / z) <= -7.851769693067612e-293) {
		tmp = t_1;
	} else if ((y / z) <= 3.176090190266049e-124) {
		tmp = t_2;
	} else if ((y / z) <= 4.2418494777197253e+219) {
		tmp = t_1;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (((y / z) * t) / t)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (z / y)
    t_2 = y * (x / z)
    if ((y / z) <= (-4.793136556035763d+249)) then
        tmp = t_2
    else if ((y / z) <= (-7.851769693067612d-293)) then
        tmp = t_1
    else if ((y / z) <= 3.176090190266049d-124) then
        tmp = t_2
    else if ((y / z) <= 4.2418494777197253d+219) then
        tmp = t_1
    else
        tmp = (1.0d0 / z) * (y * x)
    end if
    code = tmp
end function

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 / (z / y);
	double t_2 = y * (x / z);
	double tmp;
	if ((y / z) <= -4.793136556035763e+249) {
		tmp = t_2;
	} else if ((y / z) <= -7.851769693067612e-293) {
		tmp = t_1;
	} else if ((y / z) <= 3.176090190266049e-124) {
		tmp = t_2;
	} else if ((y / z) <= 4.2418494777197253e+219) {
		tmp = t_1;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x / (z / y)
	t_2 = y * (x / z)
	tmp = 0
	if (y / z) <= -4.793136556035763e+249:
		tmp = t_2
	elif (y / z) <= -7.851769693067612e-293:
		tmp = t_1
	elif (y / z) <= 3.176090190266049e-124:
		tmp = t_2
	elif (y / z) <= 4.2418494777197253e+219:
		tmp = t_1
	else:
		tmp = (1.0 / z) * (y * x)
	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(z / y))
	t_2 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= -4.793136556035763e+249)
		tmp = t_2;
	elseif (Float64(y / z) <= -7.851769693067612e-293)
		tmp = t_1;
	elseif (Float64(y / z) <= 3.176090190266049e-124)
		tmp = t_2;
	elseif (Float64(y / z) <= 4.2418494777197253e+219)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / z) * Float64(y * x));
	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 / (z / y);
	t_2 = y * (x / z);
	tmp = 0.0;
	if ((y / z) <= -4.793136556035763e+249)
		tmp = t_2;
	elseif ((y / z) <= -7.851769693067612e-293)
		tmp = t_1;
	elseif ((y / z) <= 3.176090190266049e-124)
		tmp = t_2;
	elseif ((y / z) <= 4.2418494777197253e+219)
		tmp = t_1;
	else
		tmp = (1.0 / z) * (y * x);
	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[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -4.793136556035763e+249], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -7.851769693067612e-293], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 3.176090190266049e-124], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 4.2418494777197253e+219], t$95$1, N[(N[(1.0 / z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.4
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -4.7931365560357632e249 or -7.8517696930676123e-293 < (/.f64 y z) < 3.17609019026604883e-124

   1. Initial program 20.8

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

   2. Simplified 14.2

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

   3. Applied egg-rr 14.8

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

   4. Applied egg-rr 0.8

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

   if -4.7931365560357632e249 < (/.f64 y z) < -7.8517696930676123e-293 or 3.17609019026604883e-124 < (/.f64 y z) < 4.2418494777197253e219

   1. Initial program 9.4

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

   if 4.2418494777197253e219 < (/.f64 y z)

   1. Initial program 42.8

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

   2. Simplified 28.4

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

   3. Applied egg-rr 25.6

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

   4. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{x}{\frac{1}{y}}} \]

   5. Applied egg-rr 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -4.793136556035763 \cdot 10^{+249}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -7.851769693067612 \cdot 10^{-293}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 3.176090190266049 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 4.2418494777197253 \cdot 10^{+219}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))