Numeric.Histogram:binBounds from Chart-1.5.3

Average Accuracy: 89.5% → 97.8%

Time: 13.8s

Precision: binary64

Cost: 8136

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+303}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))))
   (if (<= t_1 -1e+303)
     (+ x (* (- y x) (/ z t)))
     (if (<= t_1 5e+171) t_1 (fma (- y x) (/ z t) x)))))

C

double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double tmp;
	if (t_1 <= -1e+303) {
		tmp = x + ((y - x) * (z / t));
	} else if (t_1 <= 5e+171) {
		tmp = t_1;
	} else {
		tmp = fma((y - x), (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(Float64(y - x) * z) / t))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	tmp = 0.0
	if (t_1 <= -1e+303)
		tmp = Float64(x + Float64(Float64(y - x) * Float64(z / t)));
	elseif (t_1 <= 5e+171)
		tmp = t_1;
	else
		tmp = fma(Float64(y - x), Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+303], N[(x + N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+171], t$95$1, N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	89.5%
Target	96.6%
Herbie	97.8%

\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -1e303

   1. Initial program 8.9%

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

   2. Simplified 99.0%

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

      [Start] 8.9	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      +-commutative [=>] 8.9	\[ \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
      associate-*r/ [<=] 99.0	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
      fma-def [=>] 99.0	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

   3. Applied egg-rr 99.0%

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

      [Start] 99.0	\[ \mathsf{fma}\left(y - x, \frac{z}{t}, x\right) \]
      fma-udef [=>] 99.0	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]

   if -1e303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 5.0000000000000004e171

   1. Initial program 98.6%

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

   if 5.0000000000000004e171 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 72.5%

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

   2. Simplified 94.4%

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

      [Start] 72.5	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      +-commutative [=>] 72.5	\[ \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
      associate-*r/ [<=] 94.4	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
      fma-def [=>] 94.4	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -1 \cdot 10^{+303}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+303} \lor \neg \left(t_1 \leq 5 \cdot 10^{+277}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	1864

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 3
Accuracy	72.9%
Cost	976

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ t_2 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	89.7%
Cost	972

\[\begin{array}{l} t_1 := x + z \cdot \frac{y - x}{t}\\ \mathbf{if}\;x \leq 5.2 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-110}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 5
Accuracy	54.7%
Cost	848

\[\begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1100000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 6
Accuracy	54.7%
Cost	848

\[\begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 80000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 7
Accuracy	54.9%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-12}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;z \leq 16500000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 8
Accuracy	58.5%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -0.000108:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	58.6%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -0.000108:\\ \;\;\;\;\frac{x}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	58.5%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -0.000108:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;x \leq -1.62 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	71.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-67} \lor \neg \left(x \leq 2.1 \cdot 10^{-84}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 12
Accuracy	87.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-76} \lor \neg \left(y \leq 9 \cdot 10^{-144}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 13
Accuracy	87.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-76} \lor \neg \left(y \leq 8.2 \cdot 10^{-137}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 14
Accuracy	85.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-80}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 15
Accuracy	91.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 16
Accuracy	59.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Accuracy	50.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))