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

Average Accuracy: 92.5% → 98.1%

Time: 14.1s

Precision: binary64

Cost: 3408

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{1 - z}\\ t_2 := \frac{y}{z} - t_1\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{-128}:\\ \;\;\;\;t_2 \cdot x\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{-317}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;t_2 \leq 10^{+281}:\\ \;\;\;\;\frac{y}{z} \cdot x - t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

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 (/ t (- 1.0 z))) (t_2 (- (/ y z) t_1)))
   (if (<= t_2 (- INFINITY))
     (* y (/ x z))
     (if (<= t_2 -4e-128)
       (* t_2 x)
       (if (<= t_2 4e-317)
         (/ (+ y t) (/ z x))
         (if (<= t_2 1e+281) (- (* (/ y z) x) (* 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_2 <= -4e-128) {
		tmp = t_2 * x;
	} else if (t_2 <= 4e-317) {
		tmp = (y + t) / (z / x);
	} else if (t_2 <= 1e+281) {
		tmp = ((y / z) * x) - (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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if (t_2 <= -4e-128) {
		tmp = t_2 * x;
	} else if (t_2 <= 4e-317) {
		tmp = (y + t) / (z / x);
	} else if (t_2 <= 1e+281) {
		tmp = ((y / z) * x) - (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 = t / (1.0 - z)
	t_2 = (y / z) - t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * (x / z)
	elif t_2 <= -4e-128:
		tmp = t_2 * x
	elif t_2 <= 4e-317:
		tmp = (y + t) / (z / x)
	elif t_2 <= 1e+281:
		tmp = ((y / z) * x) - (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(t / Float64(1.0 - z))
	t_2 = Float64(Float64(y / z) - t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (t_2 <= -4e-128)
		tmp = Float64(t_2 * x);
	elseif (t_2 <= 4e-317)
		tmp = Float64(Float64(y + t) / Float64(z / x));
	elseif (t_2 <= 1e+281)
		tmp = Float64(Float64(Float64(y / z) * x) - 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 = t / (1.0 - z);
	t_2 = (y / z) - t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * (x / z);
	elseif (t_2 <= -4e-128)
		tmp = t_2 * x;
	elseif (t_2 <= 4e-317)
		tmp = (y + t) / (z / x);
	elseif (t_2 <= 1e+281)
		tmp = ((y / z) * x) - (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[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-128], N[(t$95$2 * x), $MachinePrecision], If[LessEqual[t$95$2, 4e-317], N[(N[(y + t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+281], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Target

Original	92.5%
Target	93.2%
Herbie	98.1%

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Simplified 0.0%

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

      [Start] 99.6	\[ \frac{y \cdot x}{z} \]
      associate-*l/ [<=] 0.0	\[ \color{blue}{\frac{y}{z} \cdot x} \]

   4. Taylor expanded in y around 0 99.6%

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

   5. Simplified 6.4%

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

      [Start] 99.6	\[ \frac{y \cdot x}{z} \]
      *-commutative [=>] 99.6	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      associate-/l* [=>] 6.4	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   6. Applied egg-rr 99.6%

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

      [Start] 6.4	\[ \frac{x}{\frac{z}{y}} \]
      associate-/r/ [=>] 99.6	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.00000000000000022e-128

   1. Initial program 99.7%

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

   if -4.00000000000000022e-128 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 3.99999993e-317

   1. Initial program 87.1%

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

   2. Taylor expanded in z around -inf 93.9%

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

   3. Simplified 93.2%

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

      [Start] 93.9	\[ -1 \cdot \frac{\left(-1 \cdot y - t\right) \cdot x}{z} \]
      mul-1-neg [=>] 93.9	\[ \color{blue}{-\frac{\left(-1 \cdot y - t\right) \cdot x}{z}} \]
      associate-/l* [=>] 93.2	\[ -\color{blue}{\frac{-1 \cdot y - t}{\frac{z}{x}}} \]
      mul-1-neg [=>] 93.2	\[ -\frac{\color{blue}{\left(-y\right)} - t}{\frac{z}{x}} \]

   if 3.99999993e-317 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1e281

   1. Initial program 99.6%

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

   2. Applied egg-rr 99.6%

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

      [Start] 99.6	\[ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
      sub-neg [=>] 99.6	\[ x \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)} \]
      distribute-rgt-in [=>] 99.6	\[ \color{blue}{\frac{y}{z} \cdot x + \left(-\frac{t}{1 - z}\right) \cdot x} \]
      +-commutative [=>] 99.6	\[ \color{blue}{\left(-\frac{t}{1 - z}\right) \cdot x + \frac{y}{z} \cdot x} \]
      distribute-neg-frac [=>] 99.6	\[ \color{blue}{\frac{-t}{1 - z}} \cdot x + \frac{y}{z} \cdot x \]

   if 1e281 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 34.7%

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

   2. Taylor expanded in y around inf 82.8%

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4 \cdot 10^{-128}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4 \cdot 10^{-317}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+281}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-317}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+281}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 2
Accuracy	55.4%
Cost	1113

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+47} \lor \neg \left(y \leq 3.1 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \end{array} \]

Alternative 3
Accuracy	67.1%
Cost	980

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -1.72 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 4
Accuracy	84.7%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	82.9%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	89.7%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	45.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -20000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 8
Accuracy	47.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -20000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 9
Accuracy	60.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.14 \cdot 10^{-11} \lor \neg \left(t \leq 6 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	64.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \lor \neg \left(t \leq 2.15 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 11
Accuracy	64.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 12
Accuracy	64.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 13
Accuracy	21.0%
Cost	256

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

Reproduce

herbie shell --seed 2023137 
(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)))))