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

Average Error: 7.4 → 0.6

Time: 7.4s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -2e-220) 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 <= -2e-220) {
		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 <= (-2d-220)) 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 <= -2e-220) {
		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 <= -2e-220:
		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 <= -2e-220)
		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 <= -2e-220)
		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, -2e-220], 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	7.4
Target	4.0
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))) < -1.99999999999999998e-220 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 -1.99999999999999998e-220 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 50.8

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

   2. Taylor expanded in z around 0 7.0

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

   3. Simplified 3.9

      \[\leadsto \color{blue}{\frac{y + x}{y} \cdot \left(-z\right)} \]
      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))): 70 points increase in error, 26 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 3.9

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

   5. Simplified 3.9

      \[\leadsto \color{blue}{\left(\frac{x}{y} + 1\right)} \cdot \left(-z\right) \]
      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 4.8

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

   7. Simplified 3.9

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
      Proof
      (*.f64 z (-.f64 -1 (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 z -1) (*.f64 z (/.f64 x y)))): 0 points increase in error, 2 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 z)) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) y))): 31 points increase in error, 22 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 (/.f64 (*.f64 z x) y)))): 0 points increase in error, 0 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 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.5
Cost	1236

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -1.2883115298385508 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8130638487756645 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.0299731780290312 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.813093789084415 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.23418092461876 \cdot 10^{+42}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t_0}\\ \end{array} \]

Alternative 2
Error	21.4
Cost	1108

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -1.2883115298385508 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8130638487756645 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.0299731780290312 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.813093789084415 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.23418092461876 \cdot 10^{+42}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	17.2
Cost	976

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.026416845257936 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.436269373995308 \cdot 10^{-72}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4870549074225883000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.730035447881641 \cdot 10^{+53}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	23.0
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -4.026416845257936 \cdot 10^{-35}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.436269373995308 \cdot 10^{-72}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 34291424242586220:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq 3.5832044277907327 \cdot 10^{+75}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	21.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.026416845257936 \cdot 10^{-35}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.5832044277907327 \cdot 10^{+75}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	27.7
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -4.026416845257936 \cdot 10^{-35}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.5832044277907327 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	42.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022318 
(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))))