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

Average Error: 4.6 → 1.4

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := \mathsf{fma}\left(\frac{y}{z}, x, \frac{-x}{\frac{1 - z}{t}}\right)\\ t_4 := t_1 - \frac{t \cdot x}{1 - z}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-165}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-218}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 - \left(t \cdot x + t \cdot \left(z \cdot x\right)\right)\\ \end{array} \]

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 x) z))
        (t_2 (- (/ y z) (/ t (- 1.0 z))))
        (t_3 (fma (/ y z) x (/ (- x) (/ (- 1.0 z) t))))
        (t_4 (- t_1 (/ (* t x) (- 1.0 z)))))
   (if (<= t_2 -2e+85)
     t_4
     (if (<= t_2 -5e-165)
       t_3
       (if (<= t_2 2e-218)
         t_4
         (if (<= t_2 5e+220) t_3 (- t_1 (+ (* t x) (* t (* 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 * x) / z;
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = fma((y / z), x, (-x / ((1.0 - z) / t)));
	double t_4 = t_1 - ((t * x) / (1.0 - z));
	double tmp;
	if (t_2 <= -2e+85) {
		tmp = t_4;
	} else if (t_2 <= -5e-165) {
		tmp = t_3;
	} else if (t_2 <= 2e-218) {
		tmp = t_4;
	} else if (t_2 <= 5e+220) {
		tmp = t_3;
	} else {
		tmp = t_1 - ((t * x) + (t * (z * x)));
	}
	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 * x) / z)
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_3 = fma(Float64(y / z), x, Float64(Float64(-x) / Float64(Float64(1.0 - z) / t)))
	t_4 = Float64(t_1 - Float64(Float64(t * x) / Float64(1.0 - z)))
	tmp = 0.0
	if (t_2 <= -2e+85)
		tmp = t_4;
	elseif (t_2 <= -5e-165)
		tmp = t_3;
	elseif (t_2 <= 2e-218)
		tmp = t_4;
	elseif (t_2 <= 5e+220)
		tmp = t_3;
	else
		tmp = Float64(t_1 - Float64(Float64(t * x) + Float64(t * Float64(z * x))));
	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[(y * x), $MachinePrecision] / z), $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[(N[(1.0 - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+85], t$95$4, If[LessEqual[t$95$2, -5e-165], t$95$3, If[LessEqual[t$95$2, 2e-218], t$95$4, If[LessEqual[t$95$2, 5e+220], t$95$3, N[(t$95$1 - N[(N[(t * x), $MachinePrecision] + N[(t * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.3
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2e85 or -4.99999999999999981e-165 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.0000000000000001e-218

   1. Initial program 8.7

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

   2. Taylor expanded in y around 0 2.8

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

   if -2e85 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.99999999999999981e-165 or 2.0000000000000001e-218 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.0000000000000002e220

   1. Initial program 0.2

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

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

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

   1. Initial program 22.0

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

   2. Taylor expanded in z around 0 4.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \frac{-x}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-218}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \frac{-x}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \left(t \cdot x + t \cdot \left(z \cdot x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))))