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

Average Error: 4.9 → 0.4

Time: 18.4s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{1 - z}\\ t_2 := \frac{y}{z} - t_1\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-169}:\\ \;\;\;\;\frac{y}{z} \cdot x - t_1 \cdot x\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+293}:\\ \;\;\;\;t_2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \end{array} \]

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 (/ t (- 1.0 z))) (t_2 (- (/ y z) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ y (/ z x))
     (if (<= t_2 -2e-169)
       (- (* (/ y z) x) (* t_1 x))
       (if (<= t_2 5e-237)
         (/ (* x (+ y t)) z)
         (if (<= t_2 1e+293) (* t_2 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / (z / x);
	} else if (t_2 <= -2e-169) {
		tmp = ((y / z) * x) - (t_1 * x);
	} else if (t_2 <= 5e-237) {
		tmp = (x * (y + t)) / z;
	} else if (t_2 <= 1e+293) {
		tmp = t_2 * 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y / (z / x);
	} else if (t_2 <= -2e-169) {
		tmp = ((y / z) * x) - (t_1 * x);
	} else if (t_2 <= 5e-237) {
		tmp = (x * (y + t)) / z;
	} else if (t_2 <= 1e+293) {
		tmp = t_2 * 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 = t / (1.0 - z)
	t_2 = (y / z) - t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y / (z / x)
	elif t_2 <= -2e-169:
		tmp = ((y / z) * x) - (t_1 * x)
	elif t_2 <= 5e-237:
		tmp = (x * (y + t)) / z
	elif t_2 <= 1e+293:
		tmp = t_2 * 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(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(z / x));
	elseif (t_2 <= -2e-169)
		tmp = Float64(Float64(Float64(y / z) * x) - Float64(t_1 * x));
	elseif (t_2 <= 5e-237)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (t_2 <= 1e+293)
		tmp = Float64(t_2 * x);
	else
		tmp = Float64(Float64(Float64(y * 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 = t / (1.0 - z);
	t_2 = (y / z) - t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y / (z / x);
	elseif (t_2 <= -2e-169)
		tmp = ((y / z) * x) - (t_1 * x);
	elseif (t_2 <= 5e-237)
		tmp = (x * (y + t)) / z;
	elseif (t_2 <= 1e+293)
		tmp = t_2 * 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[(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[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-169], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-237], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e+293], N[(t$95$2 * x), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] - N[(t * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	4.9
Target	4.6
Herbie	0.4

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

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 0.3

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

   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)): 48 points increase in error, 46 points decrease in error

   4. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2.00000000000000004e-169

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   if -2.00000000000000004e-169 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.0000000000000002e-237

   1. Initial program 8.5

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

   2. Taylor expanded in z around inf 9.6

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

   3. Simplified 9.6

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

   4. Taylor expanded in x around 0 1.9

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

   if 5.0000000000000002e-237 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 9.9999999999999992e292

   1. Initial program 0.2

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

   if 9.9999999999999992e292 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 50.3

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

   2. Taylor expanded in z around 0 0.4

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} + -1 \cdot \left(t \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2 \cdot 10^{-169}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+293}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+293}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \end{array} \]

Alternative 2
Error	19.9
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -8.546084017020415 \cdot 10^{+246}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -2.6856356126607676 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 4.490349989749682 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.687914798892081 \cdot 10^{+199}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 3
Error	23.4
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -2.134395281540861 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -1.430192441014468 \cdot 10^{+112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.523784740934301:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 25032142665626.715:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.6
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -2.134395281540861 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -1.430192441014468 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.523784740934301:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 25032142665626.715:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 5
Error	23.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;t \leq -2.134395281540861 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -1.430192441014468 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq -5.523784740934301:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 25032142665626.715:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 6
Error	5.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -10000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1000000:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 7
Error	3.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -10000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 1000000:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 8
Error	5.9
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y + t}}\\ \mathbf{if}\;z \leq -10000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-13}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -10000:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 10
Error	34.0
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 215000:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	34.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 215000:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 12
Error	23.7
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;t \leq -5.523784740934301:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 800481437194.3744:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	50.8
Cost	256

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

Reproduce

herbie shell --seed 2022318 
(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)))))