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

Average Error: 14.9 → 0.9

Time: 1.1min

Precision: binary64

Cost: 1492

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \begin{array}{l} \mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ t_3 := \begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+293}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+262}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

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))
        (t_2 (if (!= y 0.0) (/ x (/ z y)) t_1))
        (t_3 (if (!= x 0.0) (/ y (/ z x)) t_1)))
   (if (<= (/ y z) -5e+293)
     t_3
     (if (<= (/ y z) -1e-254)
       t_2
       (if (<= (/ y z) 2e-84)
         t_3
         (if (<= (/ y z) 5e+262) t_2 (* (/ x z) y)))))))

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 != 0.0) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	double t_2 = tmp;
	double tmp_1;
	if (x != 0.0) {
		tmp_1 = y / (z / x);
	} else {
		tmp_1 = t_1;
	}
	double t_3 = tmp_1;
	double tmp_2;
	if ((y / z) <= -5e+293) {
		tmp_2 = t_3;
	} else if ((y / z) <= -1e-254) {
		tmp_2 = t_2;
	} else if ((y / z) <= 2e-84) {
		tmp_2 = t_3;
	} else if ((y / z) <= 5e+262) {
		tmp_2 = t_2;
	} else {
		tmp_2 = (x / z) * y;
	}
	return tmp_2;
}

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) :: t_3
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_1 = (y * x) / z
    if (y /= 0.0d0) then
        tmp = x / (z / y)
    else
        tmp = t_1
    end if
    t_2 = tmp
    if (x /= 0.0d0) then
        tmp_1 = y / (z / x)
    else
        tmp_1 = t_1
    end if
    t_3 = tmp_1
    if ((y / z) <= (-5d+293)) then
        tmp_2 = t_3
    else if ((y / z) <= (-1d-254)) then
        tmp_2 = t_2
    else if ((y / z) <= 2d-84) then
        tmp_2 = t_3
    else if ((y / z) <= 5d+262) then
        tmp_2 = t_2
    else
        tmp_2 = (x / z) * y
    end if
    code = tmp_2
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 != 0.0) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	double t_2 = tmp;
	double tmp_1;
	if (x != 0.0) {
		tmp_1 = y / (z / x);
	} else {
		tmp_1 = t_1;
	}
	double t_3 = tmp_1;
	double tmp_2;
	if ((y / z) <= -5e+293) {
		tmp_2 = t_3;
	} else if ((y / z) <= -1e-254) {
		tmp_2 = t_2;
	} else if ((y / z) <= 2e-84) {
		tmp_2 = t_3;
	} else if ((y / z) <= 5e+262) {
		tmp_2 = t_2;
	} else {
		tmp_2 = (x / z) * y;
	}
	return tmp_2;
}

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 != 0.0:
		tmp = x / (z / y)
	else:
		tmp = t_1
	t_2 = tmp
	tmp_1 = 0
	if x != 0.0:
		tmp_1 = y / (z / x)
	else:
		tmp_1 = t_1
	t_3 = tmp_1
	tmp_2 = 0
	if (y / z) <= -5e+293:
		tmp_2 = t_3
	elif (y / z) <= -1e-254:
		tmp_2 = t_2
	elif (y / z) <= 2e-84:
		tmp_2 = t_3
	elif (y / z) <= 5e+262:
		tmp_2 = t_2
	else:
		tmp_2 = (x / z) * y
	return tmp_2

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(Float64(y * x) / z)
	tmp = 0.0
	if (y != 0.0)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = t_1;
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (x != 0.0)
		tmp_1 = Float64(y / Float64(z / x));
	else
		tmp_1 = t_1;
	end
	t_3 = tmp_1
	tmp_2 = 0.0
	if (Float64(y / z) <= -5e+293)
		tmp_2 = t_3;
	elseif (Float64(y / z) <= -1e-254)
		tmp_2 = t_2;
	elseif (Float64(y / z) <= 2e-84)
		tmp_2 = t_3;
	elseif (Float64(y / z) <= 5e+262)
		tmp_2 = t_2;
	else
		tmp_2 = Float64(Float64(x / z) * y);
	end
	return tmp_2
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (((y / z) * t) / t);
end

↓

function tmp_4 = code(x, y, z, t)
	t_1 = (y * x) / z;
	tmp = 0.0;
	if (y ~= 0.0)
		tmp = x / (z / y);
	else
		tmp = t_1;
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (x ~= 0.0)
		tmp_2 = y / (z / x);
	else
		tmp_2 = t_1;
	end
	t_3 = tmp_2;
	tmp_3 = 0.0;
	if ((y / z) <= -5e+293)
		tmp_3 = t_3;
	elseif ((y / z) <= -1e-254)
		tmp_3 = t_2;
	elseif ((y / z) <= 2e-84)
		tmp_3 = t_3;
	elseif ((y / z) <= 5e+262)
		tmp_3 = t_2;
	else
		tmp_3 = (x / z) * y;
	end
	tmp_4 = tmp_3;
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[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = If[Unequal[y, 0.0], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$1]}, Block[{t$95$3 = If[Unequal[x, 0.0], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], t$95$1]}, If[LessEqual[N[(y / z), $MachinePrecision], -5e+293], t$95$3, If[LessEqual[N[(y / z), $MachinePrecision], -1e-254], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 2e-84], t$95$3, If[LessEqual[N[(y / z), $MachinePrecision], 5e+262], t$95$2, N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]]]]]]

Target

Original	14.9
Target	1.5
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -5.00000000000000033e293 or -9.9999999999999991e-255 < (/.f64 y z) < 2.0000000000000001e-84

   1. Initial program 19.8

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

   2. Simplified 21.4

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

   3. Taylor expanded in x around 0 1.6

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

   4. Applied egg-rr 2.0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ } \end{array}} \]

   5. Simplified 2.0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ } \end{array}} \]
      Proof

   if -5.00000000000000033e293 < (/.f64 y z) < -9.9999999999999991e-255 or 2.0000000000000001e-84 < (/.f64 y z) < 5.00000000000000008e262

   1. Initial program 9.5

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

   2. Simplified 18.3

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

   3. Taylor expanded in x around 0 8.3

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

   4. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ } \end{array}} \]

   5. Simplified 0.2

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ } \end{array}} \]
      Proof

   if 5.00000000000000008e262 < (/.f64 y z)

   1. Initial program 54.6

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

   2. Simplified 50.6

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

   3. Taylor expanded in x around 0 0.7

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

   4. Applied egg-rr 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	1360

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := \frac{x}{z} \cdot y\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1360

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := \frac{x}{z} \cdot y\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2023033 
(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)))