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

Average Error: 7.7 → 0.3

Time: 5.8s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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-259) t_0 (if (<= t_0 0.0) (* z (- -1.0 (/ x 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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-259) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} 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-259)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    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-259) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} 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-259:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	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-259)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	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-259)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	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-259], t$95$0, If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	7.7
Target	4.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.0000000000000001e-259 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 -1.0000000000000001e-259 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 55.7

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

   2. Taylor expanded in z around 0 3.5

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

   3. Simplified 1.3

      \[\leadsto \color{blue}{\frac{y + x}{y} \cdot \left(-z\right)} \]
      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)))): 63 points increase in error, 14 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))): 65 points increase in error, 57 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

   4. Taylor expanded in y around 0 1.3

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

   5. Simplified 1.3

      \[\leadsto \color{blue}{\left(\frac{x}{y} + 1\right)} \cdot \left(-z\right) \]
      Proof
      (+.f64 (/.f64 x y) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 x y))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in x around 0 1.9

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

   7. Simplified 1.3

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
      Proof
      (*.f64 z (-.f64 -1 (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 x y) z))): 3 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (Rewrite<= *-commutative_binary64 (*.f64 z (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 (*.f64 z (/.f64 x y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 z) (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) y)))): 29 points increase in error, 13 points decrease in error
      (+.f64 (*.f64 -1 z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 z x) y)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-259}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.5
Cost	1240

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_1 := 1 - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+190}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{t_1}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 2
Error	18.4
Cost	1240

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_1 := 1 - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{+188}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{t_1}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+35}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \frac{1}{t_1}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	19.0
Cost	1108

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+36}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -2450000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.1
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+80}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	27.7
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+155}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+41}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 90000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	21.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+155}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+80}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	39.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+41}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8
Error	41.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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))))