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

Average Error: 4.8 → 1.1

Time: 10.8s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-184}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+301}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

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 (- (/ y z) (/ t (- 1.0 z)))) (t_2 (* t_1 x)))
   (if (<= t_1 (- INFINITY))
     (* y (/ x z))
     (if (<= t_1 -5e-122)
       t_2
       (if (<= t_1 1e-184)
         (/ (+ y t) (/ z x))
         (if (<= t_1 1e+301) t_2 (/ (* 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 = (y / z) - (t / (1.0 - z));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_1 <= -5e-122) {
		tmp = t_2;
	} else if (t_1 <= 1e-184) {
		tmp = (y + t) / (z / x);
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} 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 = (y / z) - (t / (1.0 - z));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if (t_1 <= -5e-122) {
		tmp = t_2;
	} else if (t_1 <= 1e-184) {
		tmp = (y + t) / (z / x);
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} 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 = (y / z) - (t / (1.0 - z))
	t_2 = t_1 * x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x / z)
	elif t_1 <= -5e-122:
		tmp = t_2
	elif t_1 <= 1e-184:
		tmp = (y + t) / (z / x)
	elif t_1 <= 1e+301:
		tmp = t_2
	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(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_2 = Float64(t_1 * x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (t_1 <= -5e-122)
		tmp = t_2;
	elseif (t_1 <= 1e-184)
		tmp = Float64(Float64(y + t) / Float64(z / x));
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	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 = (y / z) - (t / (1.0 - z));
	t_2 = t_1 * x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x / z);
	elseif (t_1 <= -5e-122)
		tmp = t_2;
	elseif (t_1 <= 1e-184)
		tmp = (y + t) / (z / x);
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	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[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-122], t$95$2, If[LessEqual[t$95$1, 1e-184], N[(N[(y + t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], t$95$2, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Target

Original	4.8
Target	4.4
Herbie	1.1

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.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)): 50 points increase in error, 53 points decrease in error

   4. Taylor expanded in y around 0 0.2

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      Proof
      (*.f64 y (/.f64 x z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y x) z)): 53 points increase in error, 51 points decrease in error

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.9999999999999999e-122 or 1.0000000000000001e-184 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.00000000000000005e301

   1. Initial program 0.2

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

   if -4.9999999999999999e-122 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.0000000000000001e-184

   1. Initial program 6.1

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

   2. Applied egg-rr 6.1

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

   3. Taylor expanded in z around inf 4.4

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

   4. Simplified 9.0

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + t}}} \]
      Proof
      (/.f64 x (/.f64 z (+.f64 y t))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 z (+.f64 y (Rewrite<= *-lft-identity_binary64 (*.f64 1 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 z (+.f64 y (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 z (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 y (*.f64 -1 t))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z)): 55 points increase in error, 68 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y (*.f64 -1 t)) x)) z): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around 0 4.4

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

   6. Simplified 4.1

      \[\leadsto \color{blue}{\frac{y + t}{\frac{z}{x}}} \]
      Proof
      (/.f64 (+.f64 y t) (/.f64 z x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 y t) x) z)): 58 points increase in error, 53 points decrease in error

   if 1.00000000000000005e301 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 57.0

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

   2. Taylor expanded in y around inf 3.1

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{-184}:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+301}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.6
Cost	1376

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := x \cdot \frac{t}{z}\\ t_3 := \frac{y}{z} \cdot x\\ t_4 := -t \cdot x\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{+183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-75}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-77}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	27.6
Cost	1376

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{y}{z} \cdot x\\ t_3 := -t \cdot x\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.7 \cdot 10^{-74}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	27.6
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ t_2 := \frac{y}{z} \cdot x\\ t_3 := -t \cdot x\\ \mathbf{if}\;z \leq -2 \cdot 10^{+275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{+190}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	27.6
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-1 - z\right)\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-71}:\\ \;\;\;\;-t \cdot x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	28.1
Cost	1244

\[\begin{array}{l} t_1 := -t \cdot x\\ t_2 := x \cdot \frac{t}{z}\\ t_3 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.42 \cdot 10^{+182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-117}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	21.2
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 7
Error	5.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \left(t + z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 8
Error	5.5
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 10
Error	28.0
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-282}:\\ \;\;\;\;-t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	50.6
Cost	256

\[-t \cdot x \]

Reproduce

herbie shell --seed 2022329 
(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)))))