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

Average Error: 4.5 → 1.2

Time: 5.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{1 - z}{t}}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - t_1\\ \mathbf{elif}\;t_2 \leq 8.332972841835484 \cdot 10^{+202}:\\ \;\;\;\;t_2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - t_1\\ \end{array} \]

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 (/ x (/ (- 1.0 z) t))) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY))
     (- (/ y (/ z x)) t_1)
     (if (<= t_2 8.332972841835484e+202) (* t_2 x) (- (* y (/ x z)) 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 = x / ((1.0 - z) / t);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / (z / x)) - t_1;
	} else if (t_2 <= 8.332972841835484e+202) {
		tmp = t_2 * x;
	} else {
		tmp = (y * (x / z)) - 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 = x / ((1.0 - z) / t);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / (z / x)) - t_1;
	} else if (t_2 <= 8.332972841835484e+202) {
		tmp = t_2 * x;
	} else {
		tmp = (y * (x / z)) - 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 = x / ((1.0 - z) / t)
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (y / (z / x)) - t_1
	elif t_2 <= 8.332972841835484e+202:
		tmp = t_2 * x
	else:
		tmp = (y * (x / z)) - 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(x / Float64(Float64(1.0 - z) / t))
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / Float64(z / x)) - t_1);
	elseif (t_2 <= 8.332972841835484e+202)
		tmp = Float64(t_2 * x);
	else
		tmp = Float64(Float64(y * Float64(x / z)) - 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 = x / ((1.0 - z) / t);
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (y / (z / x)) - t_1;
	elseif (t_2 <= 8.332972841835484e+202)
		tmp = t_2 * x;
	else
		tmp = (y * (x / z)) - 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[(x / N[(N[(1.0 - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 8.332972841835484e+202], N[(t$95$2 * x), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.3
Herbie	1.2

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

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 0 0.3

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 8.3329728418354843e202

   1. Initial program 1.3

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

   if 8.3329728418354843e202 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 20.5

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

   2. Taylor expanded in y around 0 1.5

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

   3. Applied egg-rr 1.7

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

   4. Applied egg-rr 1.2

      \[\leadsto \color{blue}{y \cdot \frac{x}{z} - \frac{x}{\frac{1 - z}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - \frac{x}{\frac{1 - z}{t}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 8.332972841835484 \cdot 10^{+202}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - \frac{x}{\frac{1 - z}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))