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

Average Error: 7.7 → 0.4

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ t_2 := \frac{x}{t_0} + \frac{y}{t_0}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(-z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))
        (t_2 (+ (/ x t_0) (/ y t_0))))
   (if (<= t_1 -5e-264) t_2 (if (<= t_1 0.0) (/ (* (+ x y) (- z)) y) t_2))))

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 t_2 = (x / t_0) + (y / t_0);
	double tmp;
	if (t_1 <= -5e-264) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = ((x + y) * -z) / y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    t_2 = (x / t_0) + (y / t_0)
    if (t_1 <= (-5d-264)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = ((x + y) * -z) / y
    else
        tmp = t_2
    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 t_2 = (x / t_0) + (y / t_0);
	double tmp;
	if (t_1 <= -5e-264) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = ((x + y) * -z) / y;
	} else {
		tmp = t_2;
	}
	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
	t_2 = (x / t_0) + (y / t_0)
	tmp = 0
	if t_1 <= -5e-264:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = ((x + y) * -z) / y
	else:
		tmp = t_2
	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)
	t_2 = Float64(Float64(x / t_0) + Float64(y / t_0))
	tmp = 0.0
	if (t_1 <= -5e-264)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(x + y) * Float64(-z)) / y);
	else
		tmp = t_2;
	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;
	t_2 = (x / t_0) + (y / t_0);
	tmp = 0.0;
	if (t_1 <= -5e-264)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = ((x + y) * -z) / y;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x / t$95$0), $MachinePrecision] + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-264], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(x + y), $MachinePrecision] * (-z)), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	3.8
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.0000000000000001e-264 or 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

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

   if -5.0000000000000001e-264 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 57.0

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

   2. Applied egg-rr 57.0

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

   3. Taylor expanded in z around 0 2.6

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

   4. Simplified 2.6

      \[\leadsto \color{blue}{\frac{\left(-z\right) \cdot \left(x + y\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(-z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))