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

Average Error: 4.6 → 1.3

Time: 5.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y}{z}\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1 - \left(x \cdot t + \left(t \cdot \left(x \cdot {z}^{2}\right) + t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq -3.6850752570351976 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 3.254386259682045 \cdot 10^{-249}:\\ \;\;\;\;t_1 - \frac{1}{\frac{1 - z}{x \cdot t}}\\ \mathbf{elif}\;t_2 \leq 5.424889879778089 \cdot 10^{+275}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - \frac{x \cdot t}{1 - z}\\ \end{array} \]

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 (/ (* x y) z)) (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (<= t_2 (- INFINITY))
     (- t_1 (+ (* x t) (+ (* t (* x (pow z 2.0))) (* t (* x z)))))
     (if (<= t_2 -3.6850752570351976e-273)
       t_2
       (if (<= t_2 3.254386259682045e-249)
         (- t_1 (/ 1.0 (/ (- 1.0 z) (* x t))))
         (if (<= t_2 5.424889879778089e+275)
           t_2
           (- (/ y (/ z x)) (/ (* x t) (- 1.0 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 = (x * y) / z;
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1 - ((x * t) + ((t * (x * pow(z, 2.0))) + (t * (x * z))));
	} else if (t_2 <= -3.6850752570351976e-273) {
		tmp = t_2;
	} else if (t_2 <= 3.254386259682045e-249) {
		tmp = t_1 - (1.0 / ((1.0 - z) / (x * t)));
	} else if (t_2 <= 5.424889879778089e+275) {
		tmp = t_2;
	} else {
		tmp = (y / (z / x)) - ((x * t) / (1.0 - 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 = (x * y) / z;
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 - ((x * t) + ((t * (x * Math.pow(z, 2.0))) + (t * (x * z))));
	} else if (t_2 <= -3.6850752570351976e-273) {
		tmp = t_2;
	} else if (t_2 <= 3.254386259682045e-249) {
		tmp = t_1 - (1.0 / ((1.0 - z) / (x * t)));
	} else if (t_2 <= 5.424889879778089e+275) {
		tmp = t_2;
	} else {
		tmp = (y / (z / x)) - ((x * t) / (1.0 - 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 = (x * y) / z
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1 - ((x * t) + ((t * (x * math.pow(z, 2.0))) + (t * (x * z))))
	elif t_2 <= -3.6850752570351976e-273:
		tmp = t_2
	elif t_2 <= 3.254386259682045e-249:
		tmp = t_1 - (1.0 / ((1.0 - z) / (x * t)))
	elif t_2 <= 5.424889879778089e+275:
		tmp = t_2
	else:
		tmp = (y / (z / x)) - ((x * t) / (1.0 - 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(x * y) / z)
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_1 - Float64(Float64(x * t) + Float64(Float64(t * Float64(x * (z ^ 2.0))) + Float64(t * Float64(x * z)))));
	elseif (t_2 <= -3.6850752570351976e-273)
		tmp = t_2;
	elseif (t_2 <= 3.254386259682045e-249)
		tmp = Float64(t_1 - Float64(1.0 / Float64(Float64(1.0 - z) / Float64(x * t))));
	elseif (t_2 <= 5.424889879778089e+275)
		tmp = t_2;
	else
		tmp = Float64(Float64(y / Float64(z / x)) - Float64(Float64(x * t) / Float64(1.0 - 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 = (x * y) / z;
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1 - ((x * t) + ((t * (x * (z ^ 2.0))) + (t * (x * z))));
	elseif (t_2 <= -3.6850752570351976e-273)
		tmp = t_2;
	elseif (t_2 <= 3.254386259682045e-249)
		tmp = t_1 - (1.0 / ((1.0 - z) / (x * t)));
	elseif (t_2 <= 5.424889879778089e+275)
		tmp = t_2;
	else
		tmp = (y / (z / x)) - ((x * t) / (1.0 - 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[(x * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$1 - N[(N[(x * t), $MachinePrecision] + N[(N[(t * N[(x * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -3.6850752570351976e-273], t$95$2, If[LessEqual[t$95$2, 3.254386259682045e-249], N[(t$95$1 - N[(1.0 / N[(N[(1.0 - z), $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5.424889879778089e+275], t$95$2, N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - N[(N[(x * t), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.3
Herbie	1.3

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x (-.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. Applied egg-rr 64.0

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

   3. Taylor expanded in z around 0 0.3

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} - \left(t \cdot x + \left(t \cdot \left({z}^{2} \cdot x\right) + t \cdot \left(z \cdot x\right)\right)\right)} \]

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < -3.68507525703519758e-273 or 3.2543862596820452e-249 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 5.424889879778089e275

   1. Initial program 0.3

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

   2. Taylor expanded in y around 0 8.0

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

   3. Taylor expanded in x around -inf 0.3

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

   if -3.68507525703519758e-273 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 3.2543862596820452e-249

   1. Initial program 6.3

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

   2. Taylor expanded in y around 0 1.9

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

   3. Applied egg-rr 1.9

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

   4. Applied egg-rr 2.9

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

   if 5.424889879778089e275 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))))

   1. Initial program 37.5

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

   2. Taylor expanded in y around 0 10.0

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

   3. Applied egg-rr 12.8

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}} + \frac{t \cdot \left(-x\right)}{1 - z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{z} - \left(x \cdot t + \left(t \cdot \left(x \cdot {z}^{2}\right) + t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -3.6850752570351976 \cdot 10^{-273}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 3.254386259682045 \cdot 10^{-249}:\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{1}{\frac{1 - z}{x \cdot t}}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 5.424889879778089 \cdot 10^{+275}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - \frac{x \cdot t}{1 - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))))