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

Average Accuracy: 88.3% → 99.7%

Time: 12.0s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-269} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-269) (not (<= t_0 0.0))) t_0 (* z (/ (- (- x) y) y)))))

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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-269) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * ((-x - y) / y);
	}
	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) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-2d-269)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-x - y) / y)
    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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-269) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * ((-x - y) / y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-269) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * ((-x - y) / y)
	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(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -2e-269) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(Float64(Float64(-x) - y) / y));
	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 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-269) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * ((-x - y) / y);
	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[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-269], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(N[((-x) - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.3%
Target	93.5%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.9999999999999999e-269 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.9999999999999999e-269 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 6.4%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Simplified 100.0%

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

      [Start] 99.8	\[ -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      mul-1-neg [=>] 99.8	\[ \color{blue}{-\frac{\left(y + x\right) \cdot z}{y}} \]
      associate-/l* [=>] 6.4	\[ -\color{blue}{\frac{y + x}{\frac{y}{z}}} \]
      +-commutative [<=] 6.4	\[ -\frac{\color{blue}{x + y}}{\frac{y}{z}} \]
      associate-/r/ [=>] 100.0	\[ -\color{blue}{\frac{x + y}{y} \cdot z} \]
      distribute-rgt-neg-in [=>] 100.0	\[ \color{blue}{\frac{x + y}{y} \cdot \left(-z\right)} \]
      +-commutative [=>] 100.0	\[ \frac{\color{blue}{y + x}}{y} \cdot \left(-z\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-269} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.4%
Cost	1304

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;y \leq 1300:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	72.7%
Cost	1304

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-81}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;y \leq 7500:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	66.5%
Cost	1241

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-94}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-116} \lor \neg \left(x \leq 4.8 \cdot 10^{-39}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	58.7%
Cost	1240

\[\begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+171}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-258}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-38}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Accuracy	70.0%
Cost	1108

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+173}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-38}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Accuracy	65.4%
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+128}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+80}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Accuracy	65.2%
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+128}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Accuracy	67.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+128}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+80}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	57.1%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+64}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10
Accuracy	41.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-161}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	35.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023153 
(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))))