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

Average Error: 14.4 → 3.2

Time: 6.0s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -4 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x z))))
   (if (<= (/ y z) -2e+236) t_1 (if (<= (/ y z) -4e-297) (* x (/ y z)) 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 = y * (x / z);
	double tmp;
	if ((y / z) <= -2e+236) {
		tmp = t_1;
	} else if ((y / z) <= -4e-297) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = y * (x / z)
    if ((y / z) <= (-2d+236)) then
        tmp = t_1
    else if ((y / z) <= (-4d-297)) then
        tmp = x * (y / z)
    else
        tmp = t_1
    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 = y * (x / z);
	double tmp;
	if ((y / z) <= -2e+236) {
		tmp = t_1;
	} else if ((y / z) <= -4e-297) {
		tmp = x * (y / z);
	} 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 = y * (x / z)
	tmp = 0
	if (y / z) <= -2e+236:
		tmp = t_1
	elif (y / z) <= -4e-297:
		tmp = x * (y / z)
	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(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= -2e+236)
		tmp = t_1;
	elseif (Float64(y / z) <= -4e-297)
		tmp = Float64(x * Float64(y / z));
	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 = y * (x / z);
	tmp = 0.0;
	if ((y / z) <= -2e+236)
		tmp = t_1;
	elseif ((y / z) <= -4e-297)
		tmp = x * (y / z);
	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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -2e+236], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -4e-297], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	14.4
Target	1.5
Herbie	3.2

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < -2.00000000000000011e236 or -4.00000000000000016e-297 < (/.f64 y z)

   1. Initial program 17.3

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

   2. Simplified 9.4

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

      [Start] 17.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 17.3	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 9.4	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 9.4	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 22.2	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 22.2	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 17.5	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 9.4	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 9.4	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 5.1

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

   4. Simplified 5.1

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

      [Start] 5.1	\[ \frac{y \cdot x}{z} \]
      rational.json-simplify-2 [<=] 5.1	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      rational.json-simplify-49 [=>] 5.1	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -2.00000000000000011e236 < (/.f64 y z) < -4.00000000000000016e-297

   1. Initial program 9.6

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

   2. Simplified 0.2

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

      [Start] 9.6	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 9.6	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 0.2	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 0.2	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 17.6	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 17.1	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 9.2	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 0.2	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 0.2	\[ x \cdot \color{blue}{\frac{y}{z}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+236}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -4 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	320

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

Reproduce

herbie shell --seed 2023073 
(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)))