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

Average Error: 7.6 → 0.1

Time: 4.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -5.852951502397782 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \end{array} \]

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 -5.852951502397782e-286)
     t_1
     (if (<= t_1 0.0) (- (fma z (/ x y) z)) (+ (/ x t_0) (/ y 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 <= -5.852951502397782e-286) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = (x / t_0) + (y / 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 <= -5.852951502397782e-286)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(Float64(x / t_0) + Float64(y / t_0));
	end
	return 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, -5.852951502397782e-286], t$95$1, If[LessEqual[t$95$1, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(N[(x / t$95$0), $MachinePrecision] + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	3.9
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.8529515023977819e-286

   1. Initial program 0.1

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.1

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

   if -5.8529515023977819e-286 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 58.9

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

   2. Taylor expanded in x around 0 58.9

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

   3. Taylor expanded in z around 0 0.3

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{-\mathsf{fma}\left(z, \frac{x}{y}, z\right)} \]

   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. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5.852951502397782 \cdot 10^{-286}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))