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

Average Error: 14.5 → 1.4

Time: 5.2s

Precision: binary64

Cost: 1360

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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) -1e+158)
     t_2
     (if (<= (/ y z) -2e-134)
       t_1
       (if (<= (/ y z) 0.0) t_2 (if (<= (/ y z) 1e+94) t_1 (* y (/ x z))))))))

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) <= -1e+158) {
		tmp = t_2;
	} else if ((y / z) <= -2e-134) {
		tmp = t_1;
	} else if ((y / z) <= 0.0) {
		tmp = t_2;
	} else if ((y / z) <= 1e+94) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	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) <= (-1d+158)) then
        tmp = t_2
    else if ((y / z) <= (-2d-134)) then
        tmp = t_1
    else if ((y / z) <= 0.0d0) then
        tmp = t_2
    else if ((y / z) <= 1d+94) then
        tmp = t_1
    else
        tmp = y * (x / z)
    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) <= -1e+158) {
		tmp = t_2;
	} else if ((y / z) <= -2e-134) {
		tmp = t_1;
	} else if ((y / z) <= 0.0) {
		tmp = t_2;
	} else if ((y / z) <= 1e+94) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	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) <= -1e+158:
		tmp = t_2
	elif (y / z) <= -2e-134:
		tmp = t_1
	elif (y / z) <= 0.0:
		tmp = t_2
	elif (y / z) <= 1e+94:
		tmp = t_1
	else:
		tmp = y * (x / z)
	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(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= -1e+158)
		tmp = t_2;
	elseif (Float64(y / z) <= -2e-134)
		tmp = t_1;
	elseif (Float64(y / z) <= 0.0)
		tmp = t_2;
	elseif (Float64(y / z) <= 1e+94)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x / z));
	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) <= -1e+158)
		tmp = t_2;
	elseif ((y / z) <= -2e-134)
		tmp = t_1;
	elseif ((y / z) <= 0.0)
		tmp = t_2;
	elseif ((y / z) <= 1e+94)
		tmp = t_1;
	else
		tmp = y * (x / z);
	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[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -1e+158], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -2e-134], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 0.0], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 1e+94], t$95$1, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	14.5
Target	1.5
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -9.99999999999999953e157 or -2.00000000000000008e-134 < (/.f64 y z) < 0.0

   1. Initial program 21.4

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

   2. Taylor expanded in x around 0 1.6

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

   if -9.99999999999999953e157 < (/.f64 y z) < -2.00000000000000008e-134 or 0.0 < (/.f64 y z) < 1e94

   1. Initial program 7.8

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
      Proof
      (/.f64 x (/.f64 z y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) z)): 43 points increase in error, 54 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 56 points increase in error, 40 points decrease in error
      (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 42 points increase in error, 2 points decrease in error

   if 1e94 < (/.f64 y z)

   1. Initial program 26.9

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

   2. Simplified 11.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
      Proof
      (/.f64 x (/.f64 z y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) z)): 43 points increase in error, 54 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 56 points increase in error, 40 points decrease in error
      (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 42 points increase in error, 2 points decrease in error

   3. Applied egg-rr 4.3

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+94}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+261}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	5.7
Cost	584

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq 4.012880369396187 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{+248}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.5
Cost	320

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

Reproduce

herbie shell --seed 2022317 
(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)))