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

Average Error: 4.4 → 2.3

Time: 11.8s

Precision: binary64

Cost: 1992

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - t \cdot x\\ \end{array} \]

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 z) (/ t (- 1.0 z)))))
   (if (<= t_1 (- INFINITY))
     (/ y (/ z x))
     (if (<= t_1 2e+164) (* t_1 x) (- (* y (/ x z)) (* t x))))))

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 / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y / (z / x);
	} else if (t_1 <= 2e+164) {
		tmp = t_1 * x;
	} else {
		tmp = (y * (x / z)) - (t * x);
	}
	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 / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y / (z / x);
	} else if (t_1 <= 2e+164) {
		tmp = t_1 * x;
	} else {
		tmp = (y * (x / z)) - (t * x);
	}
	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 / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y / (z / x)
	elif t_1 <= 2e+164:
		tmp = t_1 * x
	else:
		tmp = (y * (x / z)) - (t * x)
	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 / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y / Float64(z / x));
	elseif (t_1 <= 2e+164)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(y * Float64(x / z)) - Float64(t * x));
	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 / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y / (z / x);
	elseif (t_1 <= 2e+164)
		tmp = t_1 * x;
	else
		tmp = (y * (x / z)) - (t * x);
	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 / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+164], N[(t$95$1 * x), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(t * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.4
Target	4.1
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Taylor expanded in y around inf 0.3

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

   3. Simplified 64.0

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

      [Start] 0.3	\[ \frac{y \cdot x}{z} \]
      associate-*l/ [<=] 64.0	\[ \color{blue}{\frac{y}{z} \cdot x} \]

   4. Applied egg-rr 0.2

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2e164

   1. Initial program 1.4

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

   if 2e164 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 14.1

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

   2. Taylor expanded in z around 0 9.0

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

   3. Simplified 21.6

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

      [Start] 9.0	\[ \frac{y \cdot x}{z} + -1 \cdot \left(t \cdot x\right) \]
      +-commutative [=>] 9.0	\[ \color{blue}{-1 \cdot \left(t \cdot x\right) + \frac{y \cdot x}{z}} \]
      associate-*r* [=>] 9.0	\[ \color{blue}{\left(-1 \cdot t\right) \cdot x} + \frac{y \cdot x}{z} \]
      mul-1-neg [=>] 9.0	\[ \color{blue}{\left(-t\right)} \cdot x + \frac{y \cdot x}{z} \]
      associate-*l/ [<=] 21.6	\[ \left(-t\right) \cdot x + \color{blue}{\frac{y}{z} \cdot x} \]

   4. Taylor expanded in t around 0 9.0

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

   5. Simplified 21.6

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

      [Start] 9.0	\[ \frac{y \cdot x}{z} + -1 \cdot \left(t \cdot x\right) \]
      *-commutative [=>] 9.0	\[ \frac{\color{blue}{x \cdot y}}{z} + -1 \cdot \left(t \cdot x\right) \]
      associate-*r/ [<=] 21.6	\[ \color{blue}{x \cdot \frac{y}{z}} + -1 \cdot \left(t \cdot x\right) \]
      mul-1-neg [=>] 21.6	\[ x \cdot \frac{y}{z} + \color{blue}{\left(-t \cdot x\right)} \]
      sub-neg [<=] 21.6	\[ \color{blue}{x \cdot \frac{y}{z} - t \cdot x} \]
      *-commutative [=>] 21.6	\[ x \cdot \frac{y}{z} - \color{blue}{x \cdot t} \]

   6. Taylor expanded in x around 0 9.0

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

   7. Simplified 8.6

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

      [Start] 9.0	\[ \frac{y \cdot x}{z} - x \cdot t \]
      *-commutative [<=] 9.0	\[ \frac{\color{blue}{x \cdot y}}{z} - x \cdot t \]
      associate-*l/ [<=] 8.6	\[ \color{blue}{\frac{x}{z} \cdot y} - x \cdot t \]
      *-commutative [=>] 8.6	\[ \color{blue}{y \cdot \frac{x}{z}} - x \cdot t \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - t \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	19.5
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+251}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1400000000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{z} \cdot x - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 3
Error	4.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1400000000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{x}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 4
Error	5.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1400000000 \lor \neg \left(z \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 5
Error	5.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1400000000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 6
Error	26.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+151} \lor \neg \left(t \leq 1.26 \cdot 10^{+141}\right):\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Error	24.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 8
Error	23.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 9
Error	23.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+127}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 10
Error	50.6
Cost	256

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

Reproduce

herbie shell --seed 2023060 
(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)))))