Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.5 → 6.9

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z}{a - z}\\ t_2 := \frac{y}{a - z}\\ t_3 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y, x + z \cdot \frac{x - t}{a - z}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_1 + \left(\mathsf{fma}\left(t, t_2 - t_1, x\right) - x \cdot t_2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ z (- a z)))
        (t_2 (/ y (- a z)))
        (t_3 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_3 -4e-296)
     (fma (/ (- t x) (- a z)) y (+ x (* z (/ (- x t) (- a z)))))
     (if (<= t_3 0.0)
       (- t (/ (* (- t x) (- y a)) z))
       (+ (* x t_1) (- (fma t (- t_2 t_1) x) (* x t_2)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = z / (a - z);
	double t_2 = y / (a - z);
	double t_3 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_3 <= -4e-296) {
		tmp = fma(((t - x) / (a - z)), y, (x + (z * ((x - t) / (a - z)))));
	} else if (t_3 <= 0.0) {
		tmp = t - (((t - x) * (y - a)) / z);
	} else {
		tmp = (x * t_1) + (fma(t, (t_2 - t_1), x) - (x * t_2));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(z / Float64(a - z))
	t_2 = Float64(y / Float64(a - z))
	t_3 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_3 <= -4e-296)
		tmp = fma(Float64(Float64(t - x) / Float64(a - z)), y, Float64(x + Float64(z * Float64(Float64(x - t) / Float64(a - z)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * Float64(y - a)) / z));
	else
		tmp = Float64(Float64(x * t_1) + Float64(fma(t, Float64(t_2 - t_1), x) - Float64(x * t_2)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e-296], N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + N[(x + N[(z * N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t - N[(N[(N[(t - x), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] + N[(N[(t * N[(t$95$2 - t$95$1), $MachinePrecision] + x), $MachinePrecision] - N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	24.5
Target	11.2
Herbie	6.9

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -4e-296

   1. Initial program 20.8

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

   2. Simplified 9.8

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

   3. Taylor expanded in y around 0 18.7

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

   4. Simplified 8.2

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

   if -4e-296 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 60.2

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

   2. Simplified 60.3

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

   3. Taylor expanded in z around -inf 1.0

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

   4. Simplified 1.0

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 21.6

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

   2. Simplified 9.9

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

   3. Taylor expanded in y around 0 19.9

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

   4. Simplified 8.6

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

   5. Taylor expanded in t around 0 13.9

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

   6. Simplified 6.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -4 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y, x + z \cdot \frac{x - t}{a - z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\mathsf{fma}\left(t, \frac{y}{a - z} - \frac{z}{a - z}, x\right) - x \cdot \frac{y}{a - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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