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

Average Error: 7.5 → 0.1

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z \cdot \left(x - z\right)}{-y} - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \end{array} \]

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 -5e-292)
     t_1
     (if (<= t_1 0.0) (- (/ (* z (- x z)) (- 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 <= -5e-292) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((z * (x - z)) / -y) - z;
	} else {
		tmp = (x / t_0) + (y / 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 <= (-5d-292)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = ((z * (x - z)) / -y) - z
    else
        tmp = (x / t_0) + (y / 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 <= -5e-292) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((z * (x - z)) / -y) - z;
	} else {
		tmp = (x / t_0) + (y / 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 <= -5e-292:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = ((z * (x - z)) / -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 <= -5e-292)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(z * Float64(x - z)) / Float64(-y)) - z);
	else
		tmp = Float64(Float64(x / t_0) + Float64(y / 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 <= -5e-292)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = ((z * (x - z)) / -y) - z;
	else
		tmp = (x / t_0) + (y / 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, -5e-292], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(z * N[(x - z), $MachinePrecision]), $MachinePrecision] / (-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	7.5
Target	3.7
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))) < -4.99999999999999981e-292

   1. Initial program 0.1

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

   if -4.99999999999999981e-292 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 59.2

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

   2. Taylor expanded in y around inf 0.1

      \[\leadsto \color{blue}{-\left(\frac{z \cdot x}{y} + \left(\frac{{z}^{2}}{y} + z\right)\right)} \]

   3. Simplified 10.1

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

   4. Taylor expanded in z around 0 0.1

      \[\leadsto \left(-z\right) - \color{blue}{\left(\frac{{z}^{2}}{y} + \frac{z \cdot x}{y}\right)} \]

   5. Simplified 0.1

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

   6. Applied egg-rr 0.1

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{z \cdot \left(x - z\right)}{-y} - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))