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

Average Error: 7.6 → 0.3

Time: 24.6s

Precision: binary64

Cost: 2248

Math

\[\frac{x + y}{1 - \frac{y}{z}} \]

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0} + \frac{x}{t_0}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -1e-254)
     t_1
     (if (<= t_1 0.0) (* z (- -1.0 (/ x y))) (+ (/ y t_0) (/ x t_0))))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

↓

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -1e-254) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (y / t_0) + (x / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    if (t_1 <= (-1d-254)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = (y / t_0) + (x / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -1e-254) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (y / t_0) + (x / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

↓

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if t_1 <= -1e-254:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = (y / t_0) + (x / t_0)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

↓

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -1e-254)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(Float64(y / t_0) + Float64(x / t_0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -1e-254)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = (y / t_0) + (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-254], t$95$1, If[LessEqual[t$95$1, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / t$95$0), $MachinePrecision] + N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.6
Target	4.3
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.9999999999999991e-255

   1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   if -9.9999999999999991e-255 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 55.7

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 55.7

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}} \]

   3. Simplified 55.7

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}} \]
      Proof

      [Start] 55.7	\[ \frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}} \]
      rational.json-simplify-1 [=>] 55.7	\[ \color{blue}{\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}} \]

   4. Taylor expanded in z around 0 1.6

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{x}{y} - 1\right)} \]

   5. Simplified 1.6

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
      Proof

      [Start] 1.6	\[ z \cdot \left(-1 \cdot \frac{x}{y} - 1\right) \]
      rational.json-simplify-2 [=>] 1.6	\[ z \cdot \left(\color{blue}{\frac{x}{y} \cdot -1} - 1\right) \]
      rational.json-simplify-9 [=>] 1.6	\[ z \cdot \left(\color{blue}{\left(-\frac{x}{y}\right)} - 1\right) \]
      rational.json-simplify-64 [=>] 1.6	\[ z \cdot \color{blue}{\left(\left(-1 - \frac{x}{y}\right) + \left(1 - 1\right)\right)} \]
      metadata-eval [=>] 1.6	\[ z \cdot \left(\left(-1 - \frac{x}{y}\right) + \color{blue}{0}\right) \]
      rational.json-simplify-4 [=>] 1.6	\[ z \cdot \color{blue}{\left(-1 - \frac{x}{y}\right)} \]

   if -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}} \]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}} \]
      Proof

      [Start] 0.1	\[ \frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}} \]
      rational.json-simplify-1 [=>] 0.1	\[ \color{blue}{\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1864

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.7
Cost	1768

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_1 := 1 - \frac{y}{z}\\ t_2 := \frac{y}{t_1}\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{-19}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{-46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+49}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+136}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.3
Cost	1636

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_1 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+74}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-22}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{t_1}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-45}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;-\frac{z \cdot \left(y + x\right)}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+50}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.4
Cost	976

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{+74}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-45}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	38.5
Cost	856

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-92}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-177}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	21.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+197}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+150}:\\ \;\;\;\;z \cdot \left(-\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{+98}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	27.1
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+96}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-81}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	20.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	41.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))