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

Average Error: 14.5 → 0.6

Time: 7.4s

Precision: binary64

Cost: 1360

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+169}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

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) -5e+149)
     t_1
     (if (<= (/ y z) -1e-303)
       (* x (/ y z))
       (if (<= (/ y z) 2e-188)
         t_1
         (if (<= (/ y z) 1e+169) (/ x (/ z y)) (/ y (/ z 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 = y * (x / z);
	double tmp;
	if ((y / z) <= -5e+149) {
		tmp = t_1;
	} else if ((y / z) <= -1e-303) {
		tmp = x * (y / z);
	} else if ((y / z) <= 2e-188) {
		tmp = t_1;
	} else if ((y / z) <= 1e+169) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / 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) :: tmp
    t_1 = y * (x / z)
    if ((y / z) <= (-5d+149)) then
        tmp = t_1
    else if ((y / z) <= (-1d-303)) then
        tmp = x * (y / z)
    else if ((y / z) <= 2d-188) then
        tmp = t_1
    else if ((y / z) <= 1d+169) then
        tmp = x / (z / y)
    else
        tmp = y / (z / 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 = y * (x / z);
	double tmp;
	if ((y / z) <= -5e+149) {
		tmp = t_1;
	} else if ((y / z) <= -1e-303) {
		tmp = x * (y / z);
	} else if ((y / z) <= 2e-188) {
		tmp = t_1;
	} else if ((y / z) <= 1e+169) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	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) <= -5e+149:
		tmp = t_1
	elif (y / z) <= -1e-303:
		tmp = x * (y / z)
	elif (y / z) <= 2e-188:
		tmp = t_1
	elif (y / z) <= 1e+169:
		tmp = x / (z / y)
	else:
		tmp = y / (z / 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(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= -5e+149)
		tmp = t_1;
	elseif (Float64(y / z) <= -1e-303)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(y / z) <= 2e-188)
		tmp = t_1;
	elseif (Float64(y / z) <= 1e+169)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(y / Float64(z / 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 = y * (x / z);
	tmp = 0.0;
	if ((y / z) <= -5e+149)
		tmp = t_1;
	elseif ((y / z) <= -1e-303)
		tmp = x * (y / z);
	elseif ((y / z) <= 2e-188)
		tmp = t_1;
	elseif ((y / z) <= 1e+169)
		tmp = x / (z / y);
	else
		tmp = y / (z / 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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -5e+149], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -1e-303], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 2e-188], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 1e+169], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	14.5
Target	1.4
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -4.9999999999999999e149 or -9.99999999999999931e-304 < (/.f64 y z) < 1.9999999999999999e-188

   1. Initial program 22.4

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

   2. Simplified 14.3

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

      [Start] 22.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 22.4	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 14.3	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 14.3	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 23.7	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 23.5	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 22.9	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 14.3	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 14.3	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 1.1

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

   4. Simplified 1.1

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

      [Start] 1.1	\[ \frac{y \cdot x}{z} \]
      rational.json-simplify-2 [<=] 1.1	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      rational.json-simplify-49 [=>] 1.1	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -4.9999999999999999e149 < (/.f64 y z) < -9.99999999999999931e-304

   1. Initial program 8.6

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

   2. Simplified 0.2

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

      [Start] 8.6	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 8.6	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      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 [=>] 16.5	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 16.0	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 8.1	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 0.2	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 0.2	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if 1.9999999999999999e-188 < (/.f64 y z) < 9.99999999999999934e168

   1. Initial program 6.8

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

   2. Simplified 16.8

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

      [Start] 6.8	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 6.8	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 0.2	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-49 [<=] 6.8	\[ x \cdot \color{blue}{\frac{t \cdot \frac{y}{z}}{t}} \]
      rational.json-simplify-2 [<=] 6.8	\[ x \cdot \frac{\color{blue}{\frac{y}{z} \cdot t}}{t} \]
      rational.json-simplify-49 [=>] 7.2	\[ x \cdot \color{blue}{\left(t \cdot \frac{\frac{y}{z}}{t}\right)} \]
      rational.json-simplify-47 [=>] 16.8	\[ x \cdot \left(t \cdot \color{blue}{\frac{y}{z \cdot t}}\right) \]

   3. Taylor expanded in x around 0 9.9

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

   4. Simplified 0.2

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

      [Start] 9.9	\[ \frac{y \cdot x}{z} \]
      rational.json-simplify-7 [<=] 9.9	\[ \frac{y \cdot x}{\color{blue}{\frac{z}{1}}} \]
      rational.json-simplify-61 [<=] 10.1	\[ \color{blue}{\frac{1}{\frac{z}{y \cdot x}}} \]
      rational.json-simplify-47 [<=] 0.6	\[ \frac{1}{\color{blue}{\frac{\frac{z}{y}}{x}}} \]
      rational.json-simplify-61 [=>] 0.2	\[ \color{blue}{\frac{x}{\frac{\frac{z}{y}}{1}}} \]
      rational.json-simplify-7 [=>] 0.2	\[ \frac{x}{\color{blue}{\frac{z}{y}}} \]

   if 9.99999999999999934e168 < (/.f64 y z)

   1. Initial program 36.5

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

   2. Simplified 20.9

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

      [Start] 36.5	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 36.5	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 20.9	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 20.9	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 36.2	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 38.2	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 36.8	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 20.9	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 20.9	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 2.0

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+149}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+169}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	1360

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+212}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.6
Cost	1360

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023068 
(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)))