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

Average Error: 7.61% → 0.96%

Time: 11.1s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := t_2 \cdot x\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-221}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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-221)
       t_3
       (if (<= t_2 2e-300)
         (* (/ x z) (+ y t))
         (if (<= t_2 2e+272) 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-221) {
		tmp = t_3;
	} else if (t_2 <= 2e-300) {
		tmp = (x / z) * (y + t);
	} else if (t_2 <= 2e+272) {
		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-221) {
		tmp = t_3;
	} else if (t_2 <= 2e-300) {
		tmp = (x / z) * (y + t);
	} else if (t_2 <= 2e+272) {
		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-221:
		tmp = t_3
	elif t_2 <= 2e-300:
		tmp = (x / z) * (y + t)
	elif t_2 <= 2e+272:
		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(y * Float64(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-221)
		tmp = t_3;
	elseif (t_2 <= 2e-300)
		tmp = Float64(Float64(x / z) * Float64(y + t));
	elseif (t_2 <= 2e+272)
		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-221)
		tmp = t_3;
	elseif (t_2 <= 2e-300)
		tmp = (x / z) * (y + t);
	elseif (t_2 <= 2e+272)
		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[(y * N[(x / z), $MachinePrecision]), $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-221], t$95$3, If[LessEqual[t$95$2, 2e-300], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+272], t$95$3, t$95$1]]]]]]]

Target

Original	7.61%
Target	6.95%
Herbie	0.96%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0 or 2.0000000000000001e272 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 76.44

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

   2. Taylor expanded in y around inf 7.6

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

   3. Simplified 83.49

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

      [Start] 7.6	\[ \frac{y \cdot x}{z} \]
      associate-*l/ [<=] 83.49	\[ \color{blue}{\frac{y}{z} \cdot x} \]

   4. Taylor expanded in y around 0 7.6

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

   5. Simplified 7.66

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

      [Start] 7.6	\[ \frac{y \cdot x}{z} \]
      associate-*r/ [<=] 7.66	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.00000000000000002e-221 or 2.00000000000000005e-300 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.0000000000000001e272

   1. Initial program 0.34

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

   if -1.00000000000000002e-221 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000005e-300

   1. Initial program 21.32

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

   2. Applied egg-rr 24.31

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

   3. Taylor expanded in z around inf 1.44

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

   4. Simplified 1.34

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

      [Start] 1.44	\[ \frac{\left(y - -1 \cdot t\right) \cdot x}{z} \]
      *-commutative [<=] 1.44	\[ \frac{\color{blue}{x \cdot \left(y - -1 \cdot t\right)}}{z} \]
      associate-/l* [=>] 25.03	\[ \color{blue}{\frac{x}{\frac{z}{y - -1 \cdot t}}} \]
      associate-/r/ [=>] 1.34	\[ \color{blue}{\frac{x}{z} \cdot \left(y - -1 \cdot t\right)} \]
      mul-1-neg [=>] 1.34	\[ \frac{x}{z} \cdot \left(y - \color{blue}{\left(-t\right)}\right) \]
      sub-neg [=>] 1.34	\[ \frac{x}{z} \cdot \color{blue}{\left(y + \left(-\left(-t\right)\right)\right)} \]
      remove-double-neg [=>] 1.34	\[ \frac{x}{z} \cdot \left(y + \color{blue}{t}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.96

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{-221}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	43%
Cost	1376

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := \frac{y}{z} \cdot x\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-45}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	31.33%
Cost	1240

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	41.27%
Cost	1112

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := \frac{y}{z} \cdot x\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	32.18%
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{+289}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	35.36%
Cost	716

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-279}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 6
Error	14.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -29000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 7
Error	8.84%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -29000000 \lor \neg \left(z \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 8
Error	8.95%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \end{array} \]

Alternative 9
Error	8.98%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{\frac{y + t}{z}}{\frac{1}{x}}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \end{array} \]

Alternative 10
Error	42.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+180} \lor \neg \left(t \leq 5.7 \cdot 10^{+187}\right):\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 11
Error	36.07%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.28 \cdot 10^{+69} \lor \neg \left(t \leq 2.9 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 12
Error	78.9%
Cost	256

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

Reproduce

herbie shell --seed 2023090 
(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)))))