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

Average Error: 7.6 → 17.2

Time: 7.4s

Precision: binary64

Cost: 1432

Math

\[\frac{x + y}{1 - \frac{y}{z}} \]

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \left(-z\right) - \frac{z}{\frac{y}{z + x}}\\ t_2 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -7.63979820341188 \cdot 10^{+32}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -3.4508915249870535 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.802661005636811 \cdot 10^{-118}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.9960048458848883 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 4.1558830340779836 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.988102605020149 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z)))
        (t_1 (- (- z) (/ z (/ y (+ z x)))))
        (t_2 (* z (- -1.0 (/ x y)))))
   (if (<= y -5.6007077076010504e+110)
     t_2
     (if (<= y -7.63979820341188e+32)
       (+ y x)
       (if (<= y -3.4508915249870535e-8)
         t_1
         (if (<= y -7.802661005636811e-118)
           (+ y x)
           (if (<= y 1.9960048458848883e-14)
             (/ x t_0)
             (if (<= y 4.1558830340779836e+130)
               t_1
               (if (<= y 7.988102605020149e+159) (/ y t_0) t_2)))))))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

↓

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z / (y / (z + x)));
	double t_2 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.6007077076010504e+110) {
		tmp = t_2;
	} else if (y <= -7.63979820341188e+32) {
		tmp = y + x;
	} else if (y <= -3.4508915249870535e-8) {
		tmp = t_1;
	} else if (y <= -7.802661005636811e-118) {
		tmp = y + x;
	} else if (y <= 1.9960048458848883e-14) {
		tmp = x / t_0;
	} else if (y <= 4.1558830340779836e+130) {
		tmp = t_1;
	} else if (y <= 7.988102605020149e+159) {
		tmp = y / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = -z - (z / (y / (z + x)))
    t_2 = z * ((-1.0d0) - (x / y))
    if (y <= (-5.6007077076010504d+110)) then
        tmp = t_2
    else if (y <= (-7.63979820341188d+32)) then
        tmp = y + x
    else if (y <= (-3.4508915249870535d-8)) then
        tmp = t_1
    else if (y <= (-7.802661005636811d-118)) then
        tmp = y + x
    else if (y <= 1.9960048458848883d-14) then
        tmp = x / t_0
    else if (y <= 4.1558830340779836d+130) then
        tmp = t_1
    else if (y <= 7.988102605020149d+159) then
        tmp = y / t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z / (y / (z + x)));
	double t_2 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.6007077076010504e+110) {
		tmp = t_2;
	} else if (y <= -7.63979820341188e+32) {
		tmp = y + x;
	} else if (y <= -3.4508915249870535e-8) {
		tmp = t_1;
	} else if (y <= -7.802661005636811e-118) {
		tmp = y + x;
	} else if (y <= 1.9960048458848883e-14) {
		tmp = x / t_0;
	} else if (y <= 4.1558830340779836e+130) {
		tmp = t_1;
	} else if (y <= 7.988102605020149e+159) {
		tmp = y / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

↓

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = -z - (z / (y / (z + x)))
	t_2 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -5.6007077076010504e+110:
		tmp = t_2
	elif y <= -7.63979820341188e+32:
		tmp = y + x
	elif y <= -3.4508915249870535e-8:
		tmp = t_1
	elif y <= -7.802661005636811e-118:
		tmp = y + x
	elif y <= 1.9960048458848883e-14:
		tmp = x / t_0
	elif y <= 4.1558830340779836e+130:
		tmp = t_1
	elif y <= 7.988102605020149e+159:
		tmp = y / t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

↓

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(-z) - Float64(z / Float64(y / Float64(z + x))))
	t_2 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -5.6007077076010504e+110)
		tmp = t_2;
	elseif (y <= -7.63979820341188e+32)
		tmp = Float64(y + x);
	elseif (y <= -3.4508915249870535e-8)
		tmp = t_1;
	elseif (y <= -7.802661005636811e-118)
		tmp = Float64(y + x);
	elseif (y <= 1.9960048458848883e-14)
		tmp = Float64(x / t_0);
	elseif (y <= 4.1558830340779836e+130)
		tmp = t_1;
	elseif (y <= 7.988102605020149e+159)
		tmp = Float64(y / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = -z - (z / (y / (z + x)));
	t_2 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -5.6007077076010504e+110)
		tmp = t_2;
	elseif (y <= -7.63979820341188e+32)
		tmp = y + x;
	elseif (y <= -3.4508915249870535e-8)
		tmp = t_1;
	elseif (y <= -7.802661005636811e-118)
		tmp = y + x;
	elseif (y <= 1.9960048458848883e-14)
		tmp = x / t_0;
	elseif (y <= 4.1558830340779836e+130)
		tmp = t_1;
	elseif (y <= 7.988102605020149e+159)
		tmp = y / t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) - N[(z / N[(y / N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6007077076010504e+110], t$95$2, If[LessEqual[y, -7.63979820341188e+32], N[(y + x), $MachinePrecision], If[LessEqual[y, -3.4508915249870535e-8], t$95$1, If[LessEqual[y, -7.802661005636811e-118], N[(y + x), $MachinePrecision], If[LessEqual[y, 1.9960048458848883e-14], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 4.1558830340779836e+130], t$95$1, If[LessEqual[y, 7.988102605020149e+159], N[(y / t$95$0), $MachinePrecision], t$95$2]]]]]]]]]]

Target

Original	7.6
Target	3.8
Herbie	17.2

\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

1. Split input into 5 regimes

2. if y < -5.60070770760105043e110 or 7.9881026050201489e159 < y

   1. Initial program 21.9

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 21.9

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

   3. Taylor expanded in z around 0 9.2

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

   4. Simplified 9.2

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

   if -5.60070770760105043e110 < y < -7.63979820341188005e32 or -3.45089152498705348e-8 < y < -7.8026610056368111e-118

   1. Initial program 3.2

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 29.6

      \[\leadsto \color{blue}{y + x} \]

   if -7.63979820341188005e32 < y < -3.45089152498705348e-8 or 1.99600484588488832e-14 < y < 4.1558830340779836e130

   1. Initial program 4.8

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 4.8

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

   3. Taylor expanded in y around -inf 28.9

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{{z}^{2} + z \cdot x}{y}} \]

   4. Simplified 28.4

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

   if -7.8026610056368111e-118 < y < 1.99600484588488832e-14

   1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.2

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

   4. Taylor expanded in x around inf 13.0

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

   if 4.1558830340779836e130 < y < 7.9881026050201489e159

   1. Initial program 14.5

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 30.4

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -7.63979820341188 \cdot 10^{+32}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -3.4508915249870535 \cdot 10^{-8}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{z + x}}\\ \mathbf{elif}\;y \leq -7.802661005636811 \cdot 10^{-118}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.9960048458848883 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.1558830340779836 \cdot 10^{+130}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{z + x}}\\ \mathbf{elif}\;y \leq 7.988102605020149 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.4
Cost	1372

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.1179039927683314 \cdot 10^{-165}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 2.5918962682129878 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.233891855844952 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1730582208.003283:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.0030510805329656 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.988102605020149 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	17.1
Cost	1372

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_1 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.393890916628806 \cdot 10^{+41}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -3.4508915249870535 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.802661005636811 \cdot 10^{-118}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.9960048458848883 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;y \leq 4.1558830340779836 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.988102605020149 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.1
Cost	1240

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.393890916628806 \cdot 10^{+41}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -3.4508915249870535 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1179039927683314 \cdot 10^{-165}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 4.1558830340779836 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.988102605020149 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	23.1
Cost	1176

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.1179039927683314 \cdot 10^{-165}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 2.5918962682129878 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.233891855844952 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1730582208.003283:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.0030510805329656 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	27.6
Cost	788

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1293675114473525 \cdot 10^{+88}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -7.298635354168091 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.4508915249870535 \cdot 10^{-8}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.4757494408785894 \cdot 10^{-64}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.233891855844952 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	21.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6007077076010504 \cdot 10^{+110}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.233891855844952 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	37.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2338370608654098 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5273075910738926 \cdot 10^{-172}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	52.1
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))