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

Average Error: 4.5 → 1.9

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{1 - z}\\ t_2 := \frac{y}{z} - t_1\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-204}:\\ \;\;\;\;t_2 \cdot x\\ \mathbf{elif}\;t_2 \leq 10^{-287}:\\ \;\;\;\;\frac{y \cdot x + t \cdot x}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, t_1 \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array} \]

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 (/ t (- 1.0 z))) (t_2 (- (/ y z) t_1)))
   (if (<= t_2 -2e-204)
     (* t_2 x)
     (if (<= t_2 1e-287)
       (/ (+ (* y x) (* t x)) z)
       (if (<= t_2 1e+306)
         (fma x (/ y z) (* t_1 (- x)))
         (/ (* 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 = t / (1.0 - z);
	double t_2 = (y / z) - t_1;
	double tmp;
	if (t_2 <= -2e-204) {
		tmp = t_2 * x;
	} else if (t_2 <= 1e-287) {
		tmp = ((y * x) + (t * x)) / z;
	} else if (t_2 <= 1e+306) {
		tmp = fma(x, (y / z), (t_1 * -x));
	} 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(t / Float64(1.0 - z))
	t_2 = Float64(Float64(y / z) - t_1)
	tmp = 0.0
	if (t_2 <= -2e-204)
		tmp = Float64(t_2 * x);
	elseif (t_2 <= 1e-287)
		tmp = Float64(Float64(Float64(y * x) + Float64(t * x)) / z);
	elseif (t_2 <= 1e+306)
		tmp = fma(x, Float64(y / z), Float64(t_1 * Float64(-x)));
	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

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[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-204], N[(t$95$2 * x), $MachinePrecision], If[LessEqual[t$95$2, 1e-287], N[(N[(N[(y * x), $MachinePrecision] + N[(t * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e+306], N[(x * N[(y / z), $MachinePrecision] + N[(t$95$1 * (-x)), $MachinePrecision]), $MachinePrecision], 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]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.3
Herbie	1.9

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2e-204

   1. Initial program 3.9

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

   2. Applied egg-rr 4.0

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

   3. Taylor expanded in x around 0 3.9

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

   if -2e-204 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.00000000000000002e-287

   1. Initial program 11.2

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

   2. Taylor expanded in z around inf 1.3

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

   if 1.00000000000000002e-287 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.00000000000000002e306

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   if 1.00000000000000002e306 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 62.5

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

   2. Applied egg-rr 0.3

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{-287}:\\ \;\;\;\;\frac{y \cdot x + t \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \frac{t}{1 - z} \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))))