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

Average Error: 4.7 → 0.9

Time: 17.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := \mathsf{fma}\left(\frac{y}{z}, x, x \cdot \frac{-t}{1 - z}\right)\\ \mathbf{if}\;t_2 \leq -1.7431220279219273 \cdot 10^{+229}:\\ \;\;\;\;\frac{\left(y \cdot \left(1 - z\right) - z \cdot t\right) \cdot x}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;t_2 \leq -8.957352737326944 \cdot 10^{-192}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 6.61383725115996 \cdot 10^{+129}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma (/ y z) x (* x (/ (- t) (- 1.0 z))))))
   (if (<= t_2 -1.7431220279219273e+229)
     (/ (* (- (* y (- 1.0 z)) (* z t)) x) (* z (- 1.0 z)))
     (if (<= t_2 -8.957352737326944e-192)
       t_3
       (if (<= t_2 0.0) t_1 (if (<= t_2 6.61383725115996e+129) 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 = fma((y / z), x, (x * (-t / (1.0 - z))));
	double tmp;
	if (t_2 <= -1.7431220279219273e+229) {
		tmp = (((y * (1.0 - z)) - (z * t)) * x) / (z * (1.0 - z));
	} else if (t_2 <= -8.957352737326944e-192) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 6.61383725115996e+129) {
		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 = fma(Float64(y / z), x, Float64(x * Float64(Float64(-t) / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 <= -1.7431220279219273e+229)
		tmp = Float64(Float64(Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t)) * x) / Float64(z * Float64(1.0 - z)));
	elseif (t_2 <= -8.957352737326944e-192)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 6.61383725115996e+129)
		tmp = t_3;
	else
		tmp = t_1;
	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[(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[(N[(y / z), $MachinePrecision] * x + N[(x * N[((-t) / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.7431220279219273e+229], N[(N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(z * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -8.957352737326944e-192], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 6.61383725115996e+129], t$95$3, t$95$1]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.3
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))) < -1.74312202792192734e229

   1. Initial program 24.3

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

   2. Applied egg-rr 3.3

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

   if -1.74312202792192734e229 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -8.9573527373269441e-192 or 0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 6.61383725115995952e129

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   if -8.9573527373269441e-192 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 0.0 or 6.61383725115995952e129 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 12.3

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

   2. Taylor expanded in y around 0 2.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.7431220279219273 \cdot 10^{+229}:\\ \;\;\;\;\frac{\left(y \cdot \left(1 - z\right) - z \cdot t\right) \cdot x}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -8.957352737326944 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x \cdot \frac{-t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 6.61383725115996 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x \cdot \frac{-t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))