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

Average Error: 4.6 → 0.5

Time: 17.2s

Precision: binary64

Cost: 3664

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + t\right) + \frac{t}{z}\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array} \]

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 (/ z x))
     (if (<= t_1 -1e-175)
       t_2
       (if (<= t_1 2e-315)
         (* (/ x z) (+ (+ y t) (/ t z)))
         (if (<= t_1 4e+304)
           t_2
           (/ (* x (- (* y (- 1.0 z)) (* z t))) (* z (- 1.0 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 / (z / x);
	} else if (t_1 <= -1e-175) {
		tmp = t_2;
	} else if (t_1 <= 2e-315) {
		tmp = (x / z) * ((y + t) + (t / z));
	} else if (t_1 <= 4e+304) {
		tmp = t_2;
	} else {
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - 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 / (z / x);
	} else if (t_1 <= -1e-175) {
		tmp = t_2;
	} else if (t_1 <= 2e-315) {
		tmp = (x / z) * ((y + t) + (t / z));
	} else if (t_1 <= 4e+304) {
		tmp = t_2;
	} else {
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - 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 / (z / x)
	elif t_1 <= -1e-175:
		tmp = t_2
	elif t_1 <= 2e-315:
		tmp = (x / z) * ((y + t) + (t / z))
	elif t_1 <= 4e+304:
		tmp = t_2
	else:
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - 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(z / x));
	elseif (t_1 <= -1e-175)
		tmp = t_2;
	elseif (t_1 <= 2e-315)
		tmp = Float64(Float64(x / z) * Float64(Float64(y + t) + Float64(t / z)));
	elseif (t_1 <= 4e+304)
		tmp = t_2;
	else
		tmp = Float64(Float64(x * Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t))) / Float64(z * Float64(1.0 - 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 / (z / x);
	elseif (t_1 <= -1e-175)
		tmp = t_2;
	elseif (t_1 <= 2e-315)
		tmp = (x / z) * ((y + t) + (t / z));
	elseif (t_1 <= 4e+304)
		tmp = t_2;
	else
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - 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[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-175], t$95$2, If[LessEqual[t$95$1, 2e-315], N[(N[(x / z), $MachinePrecision] * N[(N[(y + t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+304], t$95$2, N[(N[(x * N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	4.6
Target	4.1
Herbie	0.5

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

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 0.3

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

   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, 57 points decrease in error

   4. Applied egg-rr 0.2

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1e-175 or 2.0000000019e-315 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 3.9999999999999998e304

   1. Initial program 0.2

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

   if -1e-175 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.0000000019e-315

   1. Initial program 10.8

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

   2. Taylor expanded in z around inf 2.0

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z} + \frac{t \cdot x}{{z}^{2}}} \]

   3. Simplified 2.7

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(y + t\right) + \frac{t}{z}\right)} \]
      Proof
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y t) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y (Rewrite<= *-lft-identity_binary64 (*.f64 1 t))) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) t)) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 y (*.f64 -1 t))) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 x z) (-.f64 y (*.f64 -1 t))) (*.f64 (/.f64 x z) (/.f64 t z)))): 2 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z)) (*.f64 (/.f64 x z) (/.f64 t z))): 43 points increase in error, 45 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x t) (*.f64 z z)))): 21 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t x)) (*.f64 z z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (/.f64 (*.f64 t x) (Rewrite<= unpow2_binary64 (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error

   if 3.9999999999999998e304 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 60.7

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

   2. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{\left(y \cdot \left(1 - z\right) - z \cdot t\right) \cdot x}{z \cdot \left(1 - z\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + t\right) + \frac{t}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + t\right) + \frac{t}{z}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	27.4
Cost	848

\[\begin{array}{l} t_1 := \frac{t}{\frac{z}{x}}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq -7.350912552107948 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.0053857133063414 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+165}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	25.5
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.350912552107948 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.0053857133063414 \cdot 10^{-75}:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 3.6114672902088654 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	25.5
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.350912552107948 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.0053857133063414 \cdot 10^{-75}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 3.6114672902088654 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	26.7
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ t_2 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.350912552107948 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.5612943953754977 \cdot 10^{-87}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 9.495500268273029 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	26.4
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.350912552107948 \cdot 10^{-32}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;t \leq -2.0053857133063414 \cdot 10^{-75}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 9.495500268273029 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.0
Cost	844

\[\begin{array}{l} t_1 := \frac{y + t}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.04 \cdot 10^{-243}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;z \leq 7500000:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.2
Cost	840

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

Alternative 9
Error	9.5
Cost	712

\[\begin{array}{l} t_1 := \frac{y + t}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	5.4
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y + t}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	5.2
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 12
Error	35.7
Cost	584

\[\begin{array}{l} t_1 := \frac{t}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7800000:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	50.6
Cost	256

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

Reproduce

herbie shell --seed 2022317 
(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)))))