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

Average Error: 7.7 → 0.6

Time: 8.8s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-227) t_0 (if (<= t_0 5e-302) (* (/ (+ x y) y) (- z)) 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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-227) {
		tmp = t_0;
	} else if (t_0 <= 5e-302) {
		tmp = ((x + y) / y) * -z;
	} else {
		tmp = 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) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-1d-227)) then
        tmp = t_0
    else if (t_0 <= 5d-302) then
        tmp = ((x + y) / y) * -z
    else
        tmp = 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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-227) {
		tmp = t_0;
	} else if (t_0 <= 5e-302) {
		tmp = ((x + y) / y) * -z;
	} else {
		tmp = t_0;
	}
	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 <= -1e-227:
		tmp = t_0
	elif t_0 <= 5e-302:
		tmp = ((x + y) / y) * -z
	else:
		tmp = 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(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -1e-227)
		tmp = t_0;
	elseif (t_0 <= 5e-302)
		tmp = Float64(Float64(Float64(x + y) / y) * Float64(-z));
	else
		tmp = 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 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -1e-227)
		tmp = t_0;
	elseif (t_0 <= 5e-302)
		tmp = ((x + y) / y) * -z;
	else
		tmp = 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[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-227], t$95$0, If[LessEqual[t$95$0, 5e-302], N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * (-z)), $MachinePrecision], t$95$0]]]

Target

Original	7.7
Target	4.1
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999945e-228 or 5.00000000000000033e-302 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

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

   if -9.99999999999999945e-228 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 5.00000000000000033e-302

   1. Initial program 51.7

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

   2. Taylor expanded in z around 0 6.2

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

   3. Simplified 3.6

      \[\leadsto \color{blue}{\frac{y + x}{y} \cdot \left(-z\right)} \]
      Proof
      (*.f64 (/.f64 (+.f64 y x) y) (neg.f64 z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (+.f64 y x) y) z))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 y x) z) y))): 65 points increase in error, 27 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (+.f64 y x) z) y))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.1
Cost	1372

\[\begin{array}{l} t_0 := \frac{y}{1 - \frac{y}{z}}\\ t_1 := z \cdot \frac{-x}{y}\\ \mathbf{if}\;y \leq -2.849063539903097 \cdot 10^{+125}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.065771440509976 \cdot 10^{-39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.7776120752671786 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.527565206319539 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.775127646986467 \cdot 10^{-61}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.3545064696623005 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	19.9
Cost	844

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -6.905954516999791 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.094719203971941 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 4.3545064696623005 \cdot 10^{+111}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	16.6
Cost	776

\[\begin{array}{l} t_0 := \frac{x + y}{y} \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.094719203971941 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -6.905954516999791 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.8822718277493532 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.8394023230775803 \cdot 10^{+96}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	29.3
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.433664263880772 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.402217559371769 \cdot 10^{-76}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.094719203971941 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	21.3
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.849063539903097 \cdot 10^{+125}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;y \leq 3.8394023230775803 \cdot 10^{+96}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	20.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.86386801027466 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.8394023230775803 \cdot 10^{+96}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	38.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8.469315997718042 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.134196896467881 \cdot 10^{-152}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	42.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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))))