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

Average Error: 4.6 → 0.5

Time: 1.6min

Precision: binary64

Cost: 10068

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := x \cdot t_1\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-260}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, \left(z - 1\right) \cdot y\right)}{\frac{\left(z - 1\right) \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{z - 1} + \frac{y}{z}\right) \cdot x\\ \end{array} \]

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 (- (/ y z) (/ t (- 1.0 z)))) (t_2 (* x t_1)))
   (if (<= t_1 (- INFINITY))
     (if (!= x 0.0) (/ y (/ z x)) (/ (* y x) z))
     (if (<= t_1 -2e-226)
       t_2
       (if (<= t_1 1e-260)
         (/ (* (+ y t) x) z)
         (if (<= t_1 5e+220)
           t_2
           (if (!= x 0.0)
             (/ (fma t z (* (- z 1.0) y)) (/ (* (- z 1.0) z) x))
             (* (+ (/ t (- z 1.0)) (/ y z)) 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 = (y / z) - (t / (1.0 - z));
	double t_2 = x * t_1;
	double tmp_1;
	if (t_1 <= -((double) INFINITY)) {
		double tmp_2;
		if (x != 0.0) {
			tmp_2 = y / (z / x);
		} else {
			tmp_2 = (y * x) / z;
		}
		tmp_1 = tmp_2;
	} else if (t_1 <= -2e-226) {
		tmp_1 = t_2;
	} else if (t_1 <= 1e-260) {
		tmp_1 = ((y + t) * x) / z;
	} else if (t_1 <= 5e+220) {
		tmp_1 = t_2;
	} else if (x != 0.0) {
		tmp_1 = fma(t, z, ((z - 1.0) * y)) / (((z - 1.0) * z) / x);
	} else {
		tmp_1 = ((t / (z - 1.0)) + (y / z)) * x;
	}
	return tmp_1;
}

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(x * t_1)
	tmp_1 = 0.0
	if (t_1 <= Float64(-Inf))
		tmp_2 = 0.0
		if (x != 0.0)
			tmp_2 = Float64(y / Float64(z / x));
		else
			tmp_2 = Float64(Float64(y * x) / z);
		end
		tmp_1 = tmp_2;
	elseif (t_1 <= -2e-226)
		tmp_1 = t_2;
	elseif (t_1 <= 1e-260)
		tmp_1 = Float64(Float64(Float64(y + t) * x) / z);
	elseif (t_1 <= 5e+220)
		tmp_1 = t_2;
	elseif (x != 0.0)
		tmp_1 = Float64(fma(t, z, Float64(Float64(z - 1.0) * y)) / Float64(Float64(Float64(z - 1.0) * z) / x));
	else
		tmp_1 = Float64(Float64(Float64(t / Float64(z - 1.0)) + Float64(y / z)) * x);
	end
	return tmp_1
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[(x * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], If[Unequal[x, 0.0], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]], If[LessEqual[t$95$1, -2e-226], t$95$2, If[LessEqual[t$95$1, 1e-260], N[(N[(N[(y + t), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+220], t$95$2, If[Unequal[x, 0.0], N[(N[(t * z + N[(N[(z - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z - 1.0), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]]]]

Target

Original	4.6
Target	4.0
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 64.0

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

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ } \end{array}} \]

   4. Simplified 0.5

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ } \end{array}} \]
      Proof

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.99999999999999984e-226 or 9.99999999999999961e-261 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.0000000000000002e220

   1. Initial program 0.2

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

   if -1.99999999999999984e-226 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 9.99999999999999961e-261

   1. Initial program 12.4

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

   2. Taylor expanded in z around inf 0.6

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

   3. Simplified 0.6

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

   if 5.0000000000000002e220 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 22.4

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

   2. Applied egg-rr 3.5

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 1 - z, \left(-z\right) \cdot t\right)}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{t}{z + -1} + \frac{y}{z}\right)\\ } \end{array}} \]

   3. Simplified 3.5

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, \left(z - 1\right) \cdot y\right)}{\frac{\left(z - 1\right) \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{z - 1} + \frac{y}{z}\right) \cdot x\\ } \end{array}} \]
      Proof
3. Recombined 4 regimes into one program.

Alternatives

Alternative 1
Error	0.5
Cost	3728

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := x \cdot t_1\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-260}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - z\right) \cdot \left(-y\right) + z \cdot t\right) \cdot x}{z \cdot \left(-1 + z\right)}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := x \cdot t_1\\ t_3 := \frac{y \cdot x}{z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-260}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+238}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 - t \cdot x\\ \end{array} \]

Alternative 3
Error	20.3
Cost	1112

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{t}{z}\\ t_3 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -150:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+182}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	26.9
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	27.0
Cost	980

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-1 - z\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	9.5
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -0.98:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-281}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 0.72:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	9.5
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 0.72:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

Alternative 8
Error	3.5
Cost	904

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{-z}\right)\\ \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.72:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	7.5
Cost	844

\[\begin{array}{l} t_1 := \frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.72:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \mathbf{elif}\;x \ne 0:\\ \;\;\;\;\frac{y + t}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 0.72:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

Alternative 11
Error	33.4
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	22.0
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	50.8
Cost	256

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

Reproduce

herbie shell --seed 2023033 
(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)))))