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

Average Error: 15.0 → 4.3

Time: 6.3s

Precision: binary64

Cost: 2252

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t_1 \leq -3 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (* (/ y z) t) t))) (t_2 (/ (* y x) z)))
   (if (<= t_1 -3e+287)
     t_2
     (if (<= t_1 -1e-196)
       (/ x (/ z y))
       (if (<= t_1 1e-56) t_2 (/ 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 = x * (((y / z) * t) / t);
	double t_2 = (y * x) / z;
	double tmp;
	if (t_1 <= -3e+287) {
		tmp = t_2;
	} else if (t_1 <= -1e-196) {
		tmp = x / (z / y);
	} else if (t_1 <= 1e-56) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x * (((y / z) * t) / t)
    t_2 = (y * x) / z
    if (t_1 <= (-3d+287)) then
        tmp = t_2
    else if (t_1 <= (-1d-196)) then
        tmp = x / (z / y)
    else if (t_1 <= 1d-56) then
        tmp = t_2
    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 = x * (((y / z) * t) / t);
	double t_2 = (y * x) / z;
	double tmp;
	if (t_1 <= -3e+287) {
		tmp = t_2;
	} else if (t_1 <= -1e-196) {
		tmp = x / (z / y);
	} else if (t_1 <= 1e-56) {
		tmp = t_2;
	} 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 = x * (((y / z) * t) / t)
	t_2 = (y * x) / z
	tmp = 0
	if t_1 <= -3e+287:
		tmp = t_2
	elif t_1 <= -1e-196:
		tmp = x / (z / y)
	elif t_1 <= 1e-56:
		tmp = t_2
	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(x * Float64(Float64(Float64(y / z) * t) / t))
	t_2 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (t_1 <= -3e+287)
		tmp = t_2;
	elseif (t_1 <= -1e-196)
		tmp = Float64(x / Float64(z / y));
	elseif (t_1 <= 1e-56)
		tmp = t_2;
	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 = x * (((y / z) * t) / t);
	t_2 = (y * x) / z;
	tmp = 0.0;
	if (t_1 <= -3e+287)
		tmp = t_2;
	elseif (t_1 <= -1e-196)
		tmp = x / (z / y);
	elseif (t_1 <= 1e-56)
		tmp = t_2;
	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[(x * N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -3e+287], t$95$2, If[LessEqual[t$95$1, -1e-196], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-56], t$95$2, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	15.0
Target	1.5
Herbie	4.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t)) < -2.9999999999999999e287 or -1e-196 < (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t)) < 1e-56

   1. Initial program 19.9

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

   2. Simplified 8.1

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

      [Start] 19.9	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 19.9	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 8.1	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 8.1	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 19.5	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 16.3	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 12.2	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 8.1	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 8.1	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 4.8

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

   if -2.9999999999999999e287 < (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t)) < -1e-196

   1. Initial program 0.8

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

   2. Simplified 0.4

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

      [Start] 0.8	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 0.8	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 0.4	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 17.7	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 22.6	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 15.9	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 0.4	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 0.4	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 0.6

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

   if 1e-56 < (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t))

   1. Initial program 17.9

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

   2. Simplified 7.7

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

      [Start] 17.9	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      rational.json-simplify-2 [=>] 17.9	\[ x \cdot \frac{\color{blue}{t \cdot \frac{y}{z}}}{t} \]
      rational.json-simplify-49 [=>] 7.7	\[ x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{t}{t}\right)} \]
      rational.json-simplify-2 [=>] 7.7	\[ x \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-54 [=>] 24.9	\[ x \cdot \color{blue}{\frac{\frac{y}{t}}{\frac{z}{t}}} \]
      rational.json-simplify-61 [=>] 25.0	\[ x \cdot \color{blue}{\frac{t}{\frac{z}{\frac{y}{t}}}} \]
      rational.json-simplify-61 [=>] 18.7	\[ x \cdot \frac{t}{\color{blue}{\frac{t}{\frac{y}{z}}}} \]
      rational.json-simplify-61 [=>] 7.7	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      rational.json-simplify-60 [=>] 7.7	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 6.6

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -3 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq 10^{-56}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	1100

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	2.2
Cost	1100

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+120}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.3
Cost	1100

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+120}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023075 
(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)))