Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Average Error: 4.4 → 1.3

Time: 15.5s

Precision: binary64

Cost: 1992

Math

\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{1}{x}}{y}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 (- INFINITY))
     (/ (/ 1.0 z) (/ (/ 1.0 x) y))
     (if (<= t_1 4e+299) (* t_1 x) (* (/ 1.0 z) (* y x))))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 / z) / ((1.0 / x) / y);
	} else if (t_1 <= 4e+299) {
		tmp = t_1 * x;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / z) / ((1.0 / x) / y);
	} else if (t_1 <= 4e+299) {
		tmp = t_1 * x;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

↓

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 / z) / ((1.0 / x) / y)
	elif t_1 <= 4e+299:
		tmp = t_1 * x
	else:
		tmp = (1.0 / z) * (y * x)
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / z) / Float64(Float64(1.0 / x) / y));
	elseif (t_1 <= 4e+299)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 / z) / ((1.0 / x) / y);
	elseif (t_1 <= 4e+299)
		tmp = t_1 * x;
	else
		tmp = (1.0 / z) * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 / z), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+299], N[(t$95$1 * x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.4
Target	4.1
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0

   1. Initial program 64.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   2. Taylor expanded in y around inf 0.2

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

   3. Simplified 64.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 48 points increase in error, 47 points decrease in error

   4. Applied egg-rr 0.3

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

   5. Applied egg-rr 0.4

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

   6. Applied egg-rr 0.4

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 4.0000000000000002e299

   1. Initial program 1.3

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   if 4.0000000000000002e299 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 55.8

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   2. Taylor expanded in y around inf 3.3

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

   3. Simplified 58.9

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 48 points increase in error, 47 points decrease in error

   4. Applied egg-rr 3.3

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

   5. Applied egg-rr 3.5

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

   6. Taylor expanded in y around 0 3.5

      \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(y \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{1}{x}}{y}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	26.0
Cost	1376

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{t \cdot x}{z}\\ t_3 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t \leq -1.1247661220067577 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.4205432899832886 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.561579533283366 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.12260720954978 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.7328353433096896 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.43303636957155 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.46104211268095 \cdot 10^{+99}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;t \leq 5.604584053423533 \cdot 10^{+148}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	26.9
Cost	1376

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.1911645478176845 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.5616513160418132 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.161630358129517 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5132992595817874 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 2.2924619679147225 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.2836649883436735 \cdot 10^{+237}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.3
Cost	1376

\[\begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ t_2 := \frac{x}{\frac{z}{y}}\\ t_3 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.1911645478176845 \cdot 10^{+211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.5616513160418132 \cdot 10^{+187}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.161630358129517 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7 \cdot 10^{+32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.5132992595817874 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 2.2924619679147225 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2836649883436735 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	27.3
Cost	1376

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := \frac{x}{\frac{z}{y}}\\ t_3 := t \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.1911645478176845 \cdot 10^{+211}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -2.5616513160418132 \cdot 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.161630358129517 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.5132992595817874 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 2.2924619679147225 \cdot 10^{+120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.2836649883436735 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	26.4
Cost	980

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := \frac{t \cdot x}{z}\\ \mathbf{if}\;y \leq -1.2754518745862784 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5856060336579215 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.854275674873967 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6732654396754735 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 10^{+198}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.4
Cost	844

\[\begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	12.5
Cost	844

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.5
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-16}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	27.6
Cost	584

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -1.0324625935449416 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.775777942254502 \cdot 10^{-236}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	27.0
Cost	584

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -1.2754518745862784 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3634988451344457 \cdot 10^{-37}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	50.6
Cost	256

\[t \cdot \left(-x\right) \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))