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

Average Error: 7.3 → 0.1

Time: 6.1s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -2e-294) t_0 (if (<= t_0 0.0) (* (/ (+ 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 <= -2e-294) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		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 <= (-2d-294)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) 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 <= -2e-294) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		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 <= -2e-294:
		tmp = t_0
	elif t_0 <= 0.0:
		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 <= -2e-294)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		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 <= -2e-294)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		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, -2e-294], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * (-z)), $MachinePrecision], t$95$0]]]

Target

Original	7.3
Target	4.2
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 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2.00000000000000003e-294 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

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

   if -2.00000000000000003e-294 < (/.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 z around 0 0.9

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{\frac{y + x}{y} \cdot \left(-z\right)} \]
      Proof
      (*.f64 (/.f64 (+.f64 y x) y) (neg.f64 z)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x y)) 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 x y) y) z))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 x y) (/.f64 y z)))): 56 points increase in error, 25 points decrease in error
      (neg.f64 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 y x)) (/.f64 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))): 53 points increase in error, 54 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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{x + y}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.2
Cost	1504

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -7.3 \cdot 10^{+40}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	16.7
Cost	1436

\[\begin{array}{l} t_0 := \frac{x + y}{y} \cdot \left(-z\right)\\ t_1 := 1 - \frac{y}{z}\\ t_2 := \frac{x}{t_1}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-201}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.0
Cost	976

\[\begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-195}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+110}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	21.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+110}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	27.0
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+38}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	37.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	41.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))