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

Average Accuracy: 91.9% → 97.4%

Time: 12.4s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x \cdot t\\ \end{array} \]

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 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (<= t_1 (- INFINITY))
     (/ (* x y) z)
     (if (<= t_1 4e+295) t_1 (- (/ y (/ z x)) (* x t))))))

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 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * y) / z;
	} else if (t_1 <= 4e+295) {
		tmp = t_1;
	} else {
		tmp = (y / (z / x)) - (x * t);
	}
	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 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * y) / z;
	} else if (t_1 <= 4e+295) {
		tmp = t_1;
	} else {
		tmp = (y / (z / x)) - (x * t);
	}
	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 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x * y) / z
	elif t_1 <= 4e+295:
		tmp = t_1
	else:
		tmp = (y / (z / x)) - (x * t)
	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(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * y) / z);
	elseif (t_1 <= 4e+295)
		tmp = t_1;
	else
		tmp = Float64(Float64(y / Float64(z / x)) - Float64(x * t));
	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 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x * y) / z;
	elseif (t_1 <= 4e+295)
		tmp = t_1;
	else
		tmp = (y / (z / x)) - (x * t);
	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[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 4e+295], t$95$1, N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	91.9%
Target	92.9%
Herbie	97.4%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.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%

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

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 3.9999999999999999e295

   1. Initial program 97.5%

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

   if 3.9999999999999999e295 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))))

   1. Initial program 18.3%

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

   2. Applied egg-rr 18.3%

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

      [Start] 18.3	\[ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
      flip3-- [=>] 5.7	\[ x \cdot \color{blue}{\frac{{\left(\frac{y}{z}\right)}^{3} - {\left(\frac{t}{1 - z}\right)}^{3}}{\frac{y}{z} \cdot \frac{y}{z} + \left(\frac{t}{1 - z} \cdot \frac{t}{1 - z} + \frac{y}{z} \cdot \frac{t}{1 - z}\right)}} \]
      associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{x \cdot \left({\left(\frac{y}{z}\right)}^{3} - {\left(\frac{t}{1 - z}\right)}^{3}\right)}{\frac{y}{z} \cdot \frac{y}{z} + \left(\frac{t}{1 - z} \cdot \frac{t}{1 - z} + \frac{y}{z} \cdot \frac{t}{1 - z}\right)}} \]
      associate-/l* [=>] 5.7	\[ \color{blue}{\frac{x}{\frac{\frac{y}{z} \cdot \frac{y}{z} + \left(\frac{t}{1 - z} \cdot \frac{t}{1 - z} + \frac{y}{z} \cdot \frac{t}{1 - z}\right)}{{\left(\frac{y}{z}\right)}^{3} - {\left(\frac{t}{1 - z}\right)}^{3}}}} \]
      *-un-lft-identity [=>] 5.7	\[ \frac{x}{\frac{\color{blue}{1 \cdot \left(\frac{y}{z} \cdot \frac{y}{z} + \left(\frac{t}{1 - z} \cdot \frac{t}{1 - z} + \frac{y}{z} \cdot \frac{t}{1 - z}\right)\right)}}{{\left(\frac{y}{z}\right)}^{3} - {\left(\frac{t}{1 - z}\right)}^{3}}} \]
      associate-/l* [=>] 5.7	\[ \frac{x}{\color{blue}{\frac{1}{\frac{{\left(\frac{y}{z}\right)}^{3} - {\left(\frac{t}{1 - z}\right)}^{3}}{\frac{y}{z} \cdot \frac{y}{z} + \left(\frac{t}{1 - z} \cdot \frac{t}{1 - z} + \frac{y}{z} \cdot \frac{t}{1 - z}\right)}}}} \]
      flip3-- [<=] 18.3	\[ \frac{x}{\frac{1}{\color{blue}{\frac{y}{z} - \frac{t}{1 - z}}}} \]

   3. Taylor expanded in z around 0 92.8%

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

   4. Simplified 92.2%

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

      [Start] 92.8	\[ \frac{y \cdot x}{z} + -1 \cdot \left(t \cdot x\right) \]
      +-commutative [=>] 92.8	\[ \color{blue}{-1 \cdot \left(t \cdot x\right) + \frac{y \cdot x}{z}} \]
      associate-*r* [=>] 92.8	\[ \color{blue}{\left(-1 \cdot t\right) \cdot x} + \frac{y \cdot x}{z} \]
      mul-1-neg [=>] 92.8	\[ \color{blue}{\left(-t\right)} \cdot x + \frac{y \cdot x}{z} \]
      associate-/l* [=>] 92.2	\[ \left(-t\right) \cdot x + \color{blue}{\frac{y}{\frac{z}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 4 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.4%
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+48} \lor \neg \left(z \leq 3.8 \cdot 10^{+34}\right) \land \left(z \leq 1.35 \cdot 10^{+60} \lor \neg \left(z \leq 2.7 \cdot 10^{+125}\right)\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	58.6%
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+49} \lor \neg \left(z \leq 7.8 \cdot 10^{+34} \lor \neg \left(z \leq 3.3 \cdot 10^{+72}\right) \land z \leq 3.1 \cdot 10^{+125}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	58.9%
Cost	849

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+72} \lor \neg \left(z \leq 2.7 \cdot 10^{+125}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4
Accuracy	58.8%
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 5
Accuracy	59.2%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 6
Accuracy	68.6%
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 7
Accuracy	90.9%
Cost	841

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

Alternative 8
Accuracy	93.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x \cdot t\\ \end{array} \]

Alternative 9
Accuracy	90.9%
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
Accuracy	44.6%
Cost	585

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

Alternative 11
Accuracy	20.5%
Cost	256

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

Reproduce

herbie shell --seed 2023135 
(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)))))