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

Average Error: 4.6 → 0.7

Time: 19.3s

Precision: binary64

Cost: 3408

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{1 - z}\\ t_2 := \frac{y}{z} - t_1\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-178}:\\ \;\;\;\;t_2 \cdot x\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(y + t\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{y}{z} \cdot x - t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

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 (/ t (- 1.0 z))) (t_2 (- (/ y z) t_1)))
   (if (<= t_2 (- INFINITY))
     (* y (/ x z))
     (if (<= t_2 -5e-178)
       (* t_2 x)
       (if (<= t_2 0.0)
         (* (/ 1.0 z) (* x (+ y t)))
         (if (<= t_2 2e+291) (- (* (/ y z) x) (* t_1 x)) (/ (* y x) 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_2 <= -5e-178) {
		tmp = t_2 * x;
	} else if (t_2 <= 0.0) {
		tmp = (1.0 / z) * (x * (y + t));
	} else if (t_2 <= 2e+291) {
		tmp = ((y / z) * x) - (t_1 * x);
	} else {
		tmp = (y * x) / 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if (t_2 <= -5e-178) {
		tmp = t_2 * x;
	} else if (t_2 <= 0.0) {
		tmp = (1.0 / z) * (x * (y + t));
	} else if (t_2 <= 2e+291) {
		tmp = ((y / z) * x) - (t_1 * x);
	} else {
		tmp = (y * x) / 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 = t / (1.0 - z)
	t_2 = (y / z) - t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * (x / z)
	elif t_2 <= -5e-178:
		tmp = t_2 * x
	elif t_2 <= 0.0:
		tmp = (1.0 / z) * (x * (y + t))
	elif t_2 <= 2e+291:
		tmp = ((y / z) * x) - (t_1 * x)
	else:
		tmp = (y * x) / 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(t / Float64(1.0 - z))
	t_2 = Float64(Float64(y / z) - t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (t_2 <= -5e-178)
		tmp = Float64(t_2 * x);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(x * Float64(y + t)));
	elseif (t_2 <= 2e+291)
		tmp = Float64(Float64(Float64(y / z) * x) - Float64(t_1 * x));
	else
		tmp = Float64(Float64(y * x) / 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 = t / (1.0 - z);
	t_2 = (y / z) - t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * (x / z);
	elseif (t_2 <= -5e-178)
		tmp = t_2 * x;
	elseif (t_2 <= 0.0)
		tmp = (1.0 / z) * (x * (y + t));
	elseif (t_2 <= 2e+291)
		tmp = ((y / z) * x) - (t_1 * x);
	else
		tmp = (y * x) / 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[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-178], N[(t$95$2 * x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+291], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Target

Original	4.6
Target	4.2
Herbie	0.7

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

Derivation

1. Split input into 5 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 0.2

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

   3. Simplified 64.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 57 points increase in error, 48 points decrease in error

   4. Applied egg-rr 0.3

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

   5. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.99999999999999976e-178

   1. Initial program 0.2

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

   if -4.99999999999999976e-178 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 0.0

   1. Initial program 11.5

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

   2. Taylor expanded in z around inf 1.3

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

   3. Applied egg-rr 1.3

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

   if 0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.9999999999999999e291

   1. Initial program 0.4

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

   2. Applied egg-rr 0.4

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

   if 1.9999999999999999e291 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 47.5

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

   2. Taylor expanded in y around inf 8.9

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

   3. Simplified 56.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 57 points increase in error, 48 points decrease in error

   4. Applied egg-rr 8.9

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

   5. Applied egg-rr 8.9

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

   6. Taylor expanded in x around 0 8.9

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-178}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(y + t\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(y + t\right)\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 2
Error	29.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{t \cdot x}{z}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -1.3245015632705188 \cdot 10^{+201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.637711109680108 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.193212998505264 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.933681578613223 \cdot 10^{+215}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	24.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t \leq -1.8443433845343757 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.0112545883503023 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -58705225.50428014:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 2.0476482126169987 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 9.390779298831271 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1209338484990553 \cdot 10^{+171}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	24.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t \leq -1.8443433845343757 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.0112545883503023 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -58705225.50428014:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 2.0476482126169987 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 9.390779298831271 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1209338484990553 \cdot 10^{+171}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t \leq -1.8443433845343757 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.0112545883503023 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -58705225.50428014:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 2.0476482126169987 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 9.390779298831271 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1209338484990553 \cdot 10^{+171}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;t \leq -1.8443433845343757 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -1.0112545883503023 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -58705225.50428014:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 2.0476482126169987 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 9.390779298831271 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.1209338484990553 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	7.2
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -10000000000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-166}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;\frac{y}{z} \cdot x - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 8
Error	16.9
Cost	976

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{-8}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	12.7
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y + t}}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.3
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x}{\frac{z}{y + t}}\\ \mathbf{if}\;z \leq -10000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	7.2
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -10000000000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 12
Error	27.9
Cost	584

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -6.251693996114013 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.6242408781872494 \cdot 10^{-215}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	25.9
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -4.556211739706499 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.1795658353649514 \cdot 10^{-218}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	50.3
Cost	256

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

Reproduce

herbie shell --seed 2022316 
(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)))))