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

Average Error: 4.6 → 0.3

Time: 11.9s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := t_2 \cdot x\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;t_2 \leq 10^{+305}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 x) z)) (t_2 (- (/ y z) (/ t (- 1.0 z)))) (t_3 (* t_2 x)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-252)
       t_3
       (if (<= t_2 2e-322)
         (/ (+ y t) (/ z x))
         (if (<= t_2 1e+305) t_3 t_1))))))

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 * x) / z;
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = t_2 * x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-252) {
		tmp = t_3;
	} else if (t_2 <= 2e-322) {
		tmp = (y + t) / (z / x);
	} else if (t_2 <= 1e+305) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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 * x) / z;
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = t_2 * x;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-252) {
		tmp = t_3;
	} else if (t_2 <= 2e-322) {
		tmp = (y + t) / (z / x);
	} else if (t_2 <= 1e+305) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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 * x) / z
	t_2 = (y / z) - (t / (1.0 - z))
	t_3 = t_2 * x
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-252:
		tmp = t_3
	elif t_2 <= 2e-322:
		tmp = (y + t) / (z / x)
	elif t_2 <= 1e+305:
		tmp = t_3
	else:
		tmp = t_1
	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 * x) / z)
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_3 = Float64(t_2 * x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-252)
		tmp = t_3;
	elseif (t_2 <= 2e-322)
		tmp = Float64(Float64(y + t) / Float64(z / x));
	elseif (t_2 <= 1e+305)
		tmp = t_3;
	else
		tmp = t_1;
	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 * x) / z;
	t_2 = (y / z) - (t / (1.0 - z));
	t_3 = t_2 * x;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-252)
		tmp = t_3;
	elseif (t_2 <= 2e-322)
		tmp = (y + t) / (z / x);
	elseif (t_2 <= 1e+305)
		tmp = t_3;
	else
		tmp = t_1;
	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 * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-252], t$95$3, If[LessEqual[t$95$2, 2e-322], N[(N[(y + t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], t$95$3, t$95$1]]]]]]]

Target

Original	4.6
Target	4.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0 or 9.9999999999999994e304 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 62.2

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

   2. Taylor expanded in y around inf 1.2

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -9.99999999999999943e-253 or 1.97626e-322 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 9.9999999999999994e304

   1. Initial program 0.2

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

   if -9.99999999999999943e-253 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.97626e-322

   1. Initial program 15.2

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

   2. Taylor expanded in z around -inf 0.7

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

   3. Simplified 0.8

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

      [Start] 0.7	\[ -1 \cdot \frac{\left(-1 \cdot y - t\right) \cdot x}{z} \]
      mul-1-neg [=>] 0.7	\[ \color{blue}{-\frac{\left(-1 \cdot y - t\right) \cdot x}{z}} \]
      associate-/l* [=>] 0.8	\[ -\color{blue}{\frac{-1 \cdot y - t}{\frac{z}{x}}} \]
      mul-1-neg [=>] 0.8	\[ -\frac{\color{blue}{\left(-y\right)} - t}{\frac{z}{x}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+305}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.2
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := x \cdot \frac{t}{z}\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-227}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1100:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	26.2
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := x \cdot \frac{t}{z}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-227}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 1100:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+197}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	26.0
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := x \cdot \frac{t}{z}\\ t_3 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 750:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	26.3
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-227}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq 6.2:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	26.6
Cost	1244

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-68}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.75 \cdot 10^{+18}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Error	26.0
Cost	1113

\[\begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ t_2 := t \cdot \left(-x\right)\\ t_3 := y \cdot \frac{x}{z}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{+25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+176}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+223} \lor \neg \left(t \leq 8.2 \cdot 10^{+244}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	20.5
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 8
Error	9.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.9 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 9
Error	5.3
Cost	713

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

Alternative 10
Error	5.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 11
Error	35.3
Cost	585

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

Alternative 12
Error	22.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+19} \lor \neg \left(t \leq 6.7 \cdot 10^{+25}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 13
Error	50.5
Cost	256

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

Reproduce

herbie shell --seed 2023018 
(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)))))