Complex division, imag part

Percentage Accurate: 70.1% → 89.2%

Time: 25.7s

Precision: binary64

Cost: 14025

• could not determine a ground truth

  1. a = 2.387501152545511e-262
  2. b = 1.2178575715387347e-86
  3. c = -1.3729119915258474e+173
  4. d = -5.053792087175491e+68

Math

\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+24} \lor \neg \left(d \leq 1.8 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

↓

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3e+24) (not (<= d 1.8e+94)))
   (- (* (/ c d) (/ b d)) (/ a d))
   (* (/ 1.0 (hypot c d)) (/ (- (* c b) (* d a)) (hypot c d)))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

↓

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e+24) || !(d <= 1.8e+94)) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

↓

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e+24) || !(d <= 1.8e+94)) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (((c * b) - (d * a)) / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

↓

def code(a, b, c, d):
	tmp = 0
	if (d <= -3e+24) or not (d <= 1.8e+94):
		tmp = ((c / d) * (b / d)) - (a / d)
	else:
		tmp = (1.0 / math.hypot(c, d)) * (((c * b) - (d * a)) / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

↓

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3e+24) || !(d <= 1.8e+94))
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

↓

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3e+24) || ~((d <= 1.8e+94)))
		tmp = ((c / d) * (b / d)) - (a / d);
	else
		tmp = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3e+24], N[Not[LessEqual[d, 1.8e+94]], $MachinePrecision]], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	70.1%
Target	99.2%
Herbie	89.2%

\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if d < -2.99999999999999995e24 or 1.79999999999999996e94 < d

   1. Initial program 47.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 96.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}} \]
      Step-by-step derivation

      [Start] 96.2%	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 96.2%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 96.2%	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 96.2%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      unpow2 [=>] 96.2%	\[ \frac{c \cdot b}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 99.9%	\[ \color{blue}{\frac{c}{d} \cdot \frac{b}{d}} - \frac{a}{d} \]

   if -2.99999999999999995e24 < d < 1.79999999999999996e94

   1. Initial program 84.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}} \]
      Step-by-step derivation

      [Start] 84.4%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 84.4%	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 84.4%	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 84.4%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 84.4%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 93.4%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+24} \lor \neg \left(d \leq 1.8 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.2%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+24} \lor \neg \left(d \leq 1.8 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Accuracy	86.7%
Cost	7628

\[\begin{array}{l} \mathbf{if}\;c \leq -12500000000000:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.6 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{c}{\frac{d}{b}} - a\right) \cdot \frac{1}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]

Alternative 3
Accuracy	86.7%
Cost	1356

\[\begin{array}{l} \mathbf{if}\;c \leq -12500000000000:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.7 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{c}{\frac{d}{b}} - a\right) \cdot \frac{1}{d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]

Alternative 4
Accuracy	80.7%
Cost	1236

\[\begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;d \leq -850000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.08 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \leq -12500000000000:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{c}{\frac{d}{b}} - a\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]

Alternative 6
Accuracy	81.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -10000000000000 \lor \neg \left(d \leq 2.45 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 7
Accuracy	69.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-71} \lor \neg \left(d \leq 4.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8
Accuracy	14.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 10^{+72}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 9
Accuracy	50.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 10
Accuracy	7.7%
Cost	192

\[\frac{a}{c} \]

Reproduce

herbie shell --seed 2023277 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))