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

Average Error: 4.7 → 0.9

Time: 16.3s

Precision: binary64

Cost: 3408

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := t_2 \cdot x\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-106}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+128}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 x) (- 1.0 z))))
        (t_2 (- (/ y z) (/ t (- 1.0 z))))
        (t_3 (* t_2 x)))
   (if (<= t_2 (- INFINITY))
     (* y (/ x z))
     (if (<= t_2 -1e-106)
       t_3
       (if (<= t_2 4e-318) t_1 (if (<= t_2 1e+128) 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) - ((t * x) / (1.0 - z));
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = t_2 * x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_2 <= -1e-106) {
		tmp = t_3;
	} else if (t_2 <= 4e-318) {
		tmp = t_1;
	} else if (t_2 <= 1e+128) {
		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) - ((t * x) / (1.0 - 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 = y * (x / z);
	} else if (t_2 <= -1e-106) {
		tmp = t_3;
	} else if (t_2 <= 4e-318) {
		tmp = t_1;
	} else if (t_2 <= 1e+128) {
		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 * x) / (1.0 - z))
	t_2 = (y / z) - (t / (1.0 - z))
	t_3 = t_2 * x
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * (x / z)
	elif t_2 <= -1e-106:
		tmp = t_3
	elif t_2 <= 4e-318:
		tmp = t_1
	elif t_2 <= 1e+128:
		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(Float64(Float64(y * x) / z) - Float64(Float64(t * x) / Float64(1.0 - 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 = Float64(y * Float64(x / z));
	elseif (t_2 <= -1e-106)
		tmp = t_3;
	elseif (t_2 <= 4e-318)
		tmp = t_1;
	elseif (t_2 <= 1e+128)
		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 * x) / (1.0 - z));
	t_2 = (y / z) - (t / (1.0 - z));
	t_3 = t_2 * x;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * (x / z);
	elseif (t_2 <= -1e-106)
		tmp = t_3;
	elseif (t_2 <= 4e-318)
		tmp = t_1;
	elseif (t_2 <= 1e+128)
		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[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] - N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $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)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-106], t$95$3, If[LessEqual[t$95$2, 4e-318], t$95$1, If[LessEqual[t$95$2, 1e+128], t$95$3, t$95$1]]]]]]]

Target

Original	4.7
Target	4.1
Herbie	0.9

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

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
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 54 points increase in error, 54 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))) < -9.99999999999999941e-107 or 3.9999999e-318 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.0000000000000001e128

   1. Initial program 0.2

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

   if -9.99999999999999941e-107 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 3.9999999e-318 or 1.0000000000000001e128 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 9.6

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

   2. Taylor expanded in y around 0 2.6

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{-106}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+128}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\frac{y \cdot x}{z} + \frac{t \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2
Error	26.7
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2475908464138372 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.739339072925837 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.366916493852051 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2451716588288383 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	26.7
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2475908464138372 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.739339072925837 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 8.366916493852051 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2451716588288383 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	27.0
Cost	980

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 2.2475908464138372 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.739339072925837 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 8.366916493852051 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.909261541697419 \cdot 10^{+262}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 5
Error	26.6
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 6.739339072925837 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 8.366916493852051 \cdot 10^{+193}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.909261541697419 \cdot 10^{+262}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 6
Error	26.7
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{elif}\;z \leq 6.739339072925837 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 8.366916493852051 \cdot 10^{+193}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.909261541697419 \cdot 10^{+262}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 7
Error	16.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2475908464138372 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.0
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6400000000000:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.3
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 10
Error	5.3
Cost	712

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

Alternative 11
Error	33.4
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -24:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6400000000000:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	22.2
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -4.103690038808335 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4533295911549222 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	50.4
Cost	256

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

Reproduce

herbie shell --seed 2022300 
(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)))))