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

Average Error: 24.5 → 10.0

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{t - x}{a - z}, y, \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\right)\\ \mathbf{if}\;a \leq -1:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \left(t_1 - \frac{z}{a - z}\right) + \left(x - \frac{x \cdot \left(y - z\right)}{a - z}\right)\\ \mathbf{elif}\;a \leq 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t_1, t - x, \frac{x - t}{-1 + \frac{a}{z}}\right)\\ \end{array} \]

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 (/ y (- a z)))
        (t_2 (fma (/ (- t x) (- a z)) y (fma a (/ (- t x) z) t))))
   (if (<= a -1.0)
     (+ x (* (/ (- y z) (- a z)) (- t x)))
     (if (<= a -1e-97)
       t_2
       (if (<= a -1e-190)
         (+ (* t (- t_1 (/ z (- a z)))) (- x (/ (* x (- y z)) (- a z))))
         (if (<= a 1e-5)
           t_2
           (+ x (fma t_1 (- t x) (/ (- x t) (+ -1.0 (/ a z)))))))))))

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 = y / (a - z);
	double t_2 = fma(((t - x) / (a - z)), y, fma(a, ((t - x) / z), t));
	double tmp;
	if (a <= -1.0) {
		tmp = x + (((y - z) / (a - z)) * (t - x));
	} else if (a <= -1e-97) {
		tmp = t_2;
	} else if (a <= -1e-190) {
		tmp = (t * (t_1 - (z / (a - z)))) + (x - ((x * (y - z)) / (a - z)));
	} else if (a <= 1e-5) {
		tmp = t_2;
	} else {
		tmp = x + fma(t_1, (t - x), ((x - t) / (-1.0 + (a / z))));
	}
	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(y / Float64(a - z))
	t_2 = fma(Float64(Float64(t - x) / Float64(a - z)), y, fma(a, Float64(Float64(t - x) / z), t))
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * Float64(t - x)));
	elseif (a <= -1e-97)
		tmp = t_2;
	elseif (a <= -1e-190)
		tmp = Float64(Float64(t * Float64(t_1 - Float64(z / Float64(a - z)))) + Float64(x - Float64(Float64(x * Float64(y - z)) / Float64(a - z))));
	elseif (a <= 1e-5)
		tmp = t_2;
	else
		tmp = Float64(x + fma(t_1, Float64(t - x), Float64(Float64(x - t) / Float64(-1.0 + Float64(a / z)))));
	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[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.0], N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1e-97], t$95$2, If[LessEqual[a, -1e-190], N[(N[(t * N[(t$95$1 - N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1e-5], t$95$2, N[(x + N[(t$95$1 * N[(t - x), $MachinePrecision] + N[(N[(x - t), $MachinePrecision] / N[(-1.0 + N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	24.5
Target	11.6
Herbie	10.0

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

Derivation

1. Split input into 4 regimes

2. if a < -1

   1. Initial program 22.1

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

   2. Simplified 7.9

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

   3. Taylor expanded in t around 0 14.2

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

   4. Simplified 6.1

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

   if -1 < a < -1.00000000000000004e-97 or -1e-190 < a < 1.00000000000000008e-5

   1. Initial program 27.5

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

   2. Simplified 22.2

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

   3. Taylor expanded in y around 0 27.6

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

   4. Simplified 17.7

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

   5. Taylor expanded in a around 0 23.9

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

   6. Simplified 13.3

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

   if -1.00000000000000004e-97 < a < -1e-190

   1. Initial program 28.4

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

   2. Simplified 26.4

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

   3. Taylor expanded in t around 0 18.8

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

   if 1.00000000000000008e-5 < a

   1. Initial program 21.7

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

   2. Simplified 8.9

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

   3. Taylor expanded in y around -inf 21.4

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

   4. Simplified 7.2

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y, \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + \left(x - \frac{x \cdot \left(y - z\right)}{a - z}\right)\\ \mathbf{elif}\;a \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y, \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{y}{a - z}, t - x, \frac{x - t}{-1 + \frac{a}{z}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022182 
(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))))