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

Average Error: 7.4 → 0.3

Time: 6.8s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0} + \frac{x}{t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -5e-267)
     t_1
     (if (<= t_1 0.0) (- (- z) (/ z (/ y (+ x z)))) (+ (/ y t_0) (/ x 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 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -z - (z / (y / (x + z)));
	} else {
		tmp = (y / t_0) + (x / 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    if (t_1 <= (-5d-267)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = -z - (z / (y / (x + z)))
    else
        tmp = (y / t_0) + (x / 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 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -z - (z / (y / (x + z)));
	} else {
		tmp = (y / t_0) + (x / t_0);
	}
	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
	tmp = 0
	if t_1 <= -5e-267:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = -z - (z / (y / (x + z)))
	else:
		tmp = (y / t_0) + (x / 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(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -5e-267)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-z) - Float64(z / Float64(y / Float64(x + z))));
	else
		tmp = Float64(Float64(y / t_0) + Float64(x / 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 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -5e-267)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = -z - (z / (y / (x + z)));
	else
		tmp = (y / t_0) + (x / 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[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-267], t$95$1, If[LessEqual[t$95$1, 0.0], N[((-z) - N[(z / N[(y / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / t$95$0), $MachinePrecision] + N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.4
Target	4.0
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 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.9999999999999999e-267

   1. Initial program 0.1

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

   if -4.9999999999999999e-267 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 56.7

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

   2. Taylor expanded in y around inf 1.6

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]

   3. Simplified 1.6

      \[\leadsto \color{blue}{\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}} \]
      Proof
      (-.f64 (neg.f64 z) (/.f64 (*.f64 z (+.f64 x z)) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) (/.f64 (*.f64 z (+.f64 x z)) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 z x) (*.f64 z z))) y)): 1 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 -1 z) (/.f64 (Rewrite<= cancel-sign-sub_binary64 (-.f64 (*.f64 z x) (*.f64 (neg.f64 z) z))) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (/.f64 (-.f64 (*.f64 z x) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 z z)))) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (/.f64 (-.f64 (*.f64 z x) (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)))) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (/.f64 (-.f64 (*.f64 z x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 z 2)))) y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 z x) y) (/.f64 (*.f64 -1 (pow.f64 z 2)) y)))): 1 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 -1 z) (-.f64 (/.f64 (*.f64 z x) y) (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 z) (-.f64 (/.f64 (*.f64 z x) y) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (pow.f64 z 2) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 (*.f64 -1 z) (/.f64 (*.f64 z x) y)) (neg.f64 (/.f64 (pow.f64 z 2) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 (/.f64 (*.f64 z x) y)))) (neg.f64 (/.f64 (pow.f64 z 2) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 z x) y)))) (neg.f64 (/.f64 (pow.f64 z 2) y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 z x) y))) (/.f64 (pow.f64 z 2) y))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 1.6

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

   5. Applied egg-rr 1.4

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

   if 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

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}} \]
      Proof
      (+.f64 (/.f64 y (-.f64 1 (/.f64 y z))) (/.f64 x (-.f64 1 (/.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x (-.f64 1 (/.f64 y z))) (/.f64 y (-.f64 1 (/.f64 y z))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1928

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x + z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1864

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.8
Cost	844

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -1.788122082435576 \cdot 10^{+120}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -3.5470999373193547 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 2.7069182788710137 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	16.9
Cost	776

\[\begin{array}{l} t_0 := \left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -2.4438914743020444 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.613914637093929 \cdot 10^{-64}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	20.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -9.567327470886502 \cdot 10^{+79}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 2.7069182788710137 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	20.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -4.395808124878851 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.4859294020564273 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	20.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.395808124878851 \cdot 10^{+95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.4859294020564273 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	27.2
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4438914743020444 \cdot 10^{-30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.37320025760238 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	38.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0209056636638623 \cdot 10^{-180}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.922498490892363 \cdot 10^{-234}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	41.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022295 
(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))))