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

Average Error: 4.6 → 4.0

Time: 23.6s

Precision: binary64

Cost: 8328

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 1 - z, -z \cdot t\right)}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \end{array} \]

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 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (<= t_1 2e+242)
     t_1
     (if (!= x 0.0)
       (/ (fma y (- 1.0 z) (- (* z t))) (/ (* z (- 1.0 z)) x))
       (* x (+ (/ y z) (/ t (+ z -1.0))))))))

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 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_1 <= 2e+242) {
		tmp = t_1;
	} else if (x != 0.0) {
		tmp = fma(y, (1.0 - z), -(z * t)) / ((z * (1.0 - z)) / x);
	} else {
		tmp = x * ((y / z) + (t / (z + -1.0)));
	}
	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(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_1 <= 2e+242)
		tmp = t_1;
	elseif (x != 0.0)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-Float64(z * t))) / Float64(Float64(z * Float64(1.0 - z)) / x));
	else
		tmp = Float64(x * Float64(Float64(y / z) + Float64(t / Float64(z + -1.0))));
	end
	return 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[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+242], t$95$1, If[Unequal[x, 0.0], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[(z * t), $MachinePrecision])), $MachinePrecision] / N[(N[(z * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.6
Target	4.3
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 2.0000000000000001e242

   1. Initial program 3.0

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

   if 2.0000000000000001e242 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))))

   1. Initial program 26.8

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

   2. Applied egg-rr 18.0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 1 - z, -z \cdot t\right)}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ } \end{array}} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	3.1
Cost	1476

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t_1 \leq 10^{+287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \end{array} \]

Alternative 2
Error	27.6
Cost	1112

\[\begin{array}{l} t_1 := x \cdot \left(-t\right)\\ t_2 := x \cdot \frac{y}{z}\\ t_3 := \frac{x}{z} \cdot y\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-226}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+161}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 3
Error	20.6
Cost	976

\[\begin{array}{l} t_1 := \frac{t}{-1 + z} \cdot x\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-188}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4
Error	24.5
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 5
Error	6.0
Cost	840

\[\begin{array}{l} t_1 := \left(\frac{t}{z} + \frac{y}{z}\right) \cdot x\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	4.2
Cost	840

\[\begin{array}{l} t_1 := \left(\frac{t}{z} + \frac{y}{z}\right) \cdot x\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.8
Cost	716

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 8
Error	10.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	6.0
Cost	712

\[\begin{array}{l} t_1 := \frac{y + t}{z} \cdot x\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.6
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	24.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 12
Error	50.5
Cost	256

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

Reproduce

herbie shell --seed 2023010 
(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)))))