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

Average Accuracy: 94.2% → 98.5%

Time: 12.1s

Precision: binary64

Cost: 1992

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

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))
     (* y (/ x z))
     (if (<= t_1 1e+307) (* t_1 x) (/ (* y x) z)))))

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 = y * (x / z);
	} else if (t_1 <= 1e+307) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	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 = y * (x / z);
	} else if (t_1 <= 1e+307) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	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 = y * (x / z)
	elif t_1 <= 1e+307:
		tmp = t_1 * x
	else:
		tmp = (y * x) / z
	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(y * Float64(x / z));
	elseif (t_1 <= 1e+307)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(y * x) / z);
	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 = y * (x / z);
	elseif (t_1 <= 1e+307)
		tmp = t_1 * x;
	else
		tmp = (y * x) / z;
	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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], N[(t$95$1 * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	94.2%
Target	94.6%
Herbie	98.5%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 70.8%

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

   2. Taylor expanded in z around 0 93.6%

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

   3. Simplified 70.8%

      \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} - t\right)} \]
      Step-by-step derivation

      [Start] 93.6	\[ \frac{y \cdot x}{z} + -1 \cdot \left(t \cdot x\right) \]
      associate-*l/ [<=] 64.5	\[ \color{blue}{\frac{y}{z} \cdot x} + -1 \cdot \left(t \cdot x\right) \]
      associate-*r* [=>] 64.5	\[ \frac{y}{z} \cdot x + \color{blue}{\left(-1 \cdot t\right) \cdot x} \]
      neg-mul-1 [<=] 64.5	\[ \frac{y}{z} \cdot x + \color{blue}{\left(-t\right)} \cdot x \]
      distribute-rgt-out [=>] 70.8	\[ \color{blue}{x \cdot \left(\frac{y}{z} + \left(-t\right)\right)} \]
      unsub-neg [=>] 70.8	\[ x \cdot \color{blue}{\left(\frac{y}{z} - t\right)} \]

   4. Taylor expanded in y around inf 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      Step-by-step derivation

      [Start] 99.8	\[ \frac{y \cdot x}{z} \]
      *-commutative [=>] 99.8	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      associate-/l* [=>] 70.8	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]
      associate-/r/ [=>] 99.8	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 9.99999999999999986e306

   1. Initial program 98.6%

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

   if 9.99999999999999986e306 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+307}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	62.6%
Cost	1113

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.16 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-287}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+29} \lor \neg \left(y \leq 1.05 \cdot 10^{+47}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	62.6%
Cost	1113

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-151}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+26} \lor \neg \left(y \leq 1.05 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	62.4%
Cost	1113

\[\begin{array}{l} t_1 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-188}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 3.95 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+29} \lor \neg \left(y \leq 1.05 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 4
Accuracy	62.3%
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-144}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 5
Accuracy	62.2%
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 6
Accuracy	62.5%
Cost	1112

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := t \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-288}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 2.52 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 7
Accuracy	88.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -21 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 8
Accuracy	93.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -21 \lor \neg \left(z \leq 1300000\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 9
Accuracy	74.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 10
Accuracy	42.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -14.2 \lor \neg \left(z \leq 1.1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 11
Accuracy	44.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -14.2 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 12
Accuracy	22.7%
Cost	256

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

Reproduce

herbie shell --seed 2023159 
(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)))))